 But here is the final part of my talk, where I'm going to now introduce to you how those topological phases that I was introducing in the last few lectures are being realized in ultracord atomic and photonic experiments today. So this is just a reminder of where we are in my overview. And now I've also added all of the main references that I think are really pedagogical introductions to artificial gauge fields and topological phases of matter for cold atomic systems. So I do recommend that you have a look at those. And my slides are online. So again, don't worry about writing all of this down. OK, so I'm a little bit unlucky in that I was meant to be talking after Professor Javier had already given us a bit of an introduction to cold atoms. Unfortunately, as he has not been able to make it yet, you will get the introduction to cold atoms after my talk. So I will very quickly go through some of the main features of cold atoms that we're really interested in using and that we see as a resource for exploring condensed matter physics in new ways. But there will be more details on much of this coming later in the week. And so one of the first things to really emphasize is that unlike the ions that we were hearing about earlier, we're talking now about collections of neutral particles, so uncharged particles. And we can trap and cool different types of atoms. So we can make bisonic gases or fermionic gases. And now a very important area of research is actually making mixtures of bosons and fermions in the same trap and then seeing what happens. And so that's already a difference to what I was talking about the last couple of lectures, whereas talking about fermions all the time, because I was putting on my condensed matter viewpoints. And so one question is, what other things could we do with bosons that we're still trying to understand in this field? Another very useful thing about cold atoms is that we have the opportunity to tune the interaction strength. And so this is through something called a Feshbach resonance, not explaining what really this is, the important thing is that as we tune, for instance, an external magnetic field, we can also tune the effective, in this case, the contact interactions between species. So even though they're not interacting via a Coulomb interaction because they're not charged, they've got this tunable contact interaction. And they can also have other types of interactions, but I'm not really talking about interacting physics, so I'm leaving this as a prospect for a cool thing that we can do in the future. Another point is that these are very controllable systems. As we see in the next few slides, we're gonna use lasers to control the different geometry of how we trap the atoms very much like we were hearing about with the ion experiments as well earlier today. Another thing that we heard about already is that the great things about using these atoms or ions is that we have addressable internal states. And in particular, the hyperfine states of atoms give us a wonderful manifold of states that we can try and play with and tune, which will be very important when we're trying to simulate artificial gauge fields. And then in terms of the observables, we can do different things than we can do in solid state physics. We can look at the atomic cloud using a time of flight imaging, which means we just release the trap and we let the cloud expand and then we image it. And because that depends on how fast those atoms are moving, that gives us a measure of what momenta the atoms had. So how far they're going in this time of flight is telling us about the momenta distribution of the atoms. This is a classic picture where you can see the emergence of the Bose-Einstein condensate peak, which is this peak in the momentum distribution. But now we can also do much more in terms of accessing the density in real space distribution of atoms. Particularly in the last few years, there's been great development in doing very, very high resolution optical imaging, using things for instance called quantum gas microscopes, which can even image with single site resolution of an optical lattice where those atoms are. And so the other thing I want to really emphasize, which again you'll hear a lot more about tomorrow, is how we can use those laser fields to create effectively potentials for light, so crystals, potentials of light, so crystals for the atoms to move in. And so the idea is that the atoms are going to be feeling this optical dipole potential, which depends on their polarizability, and also on the intensity of the light that you are applying. And so if you interfere light to make different standing interference patterns, that interference pattern will be felt by the atoms as a potential that they are moving in. And we can do this in 1D, 2D, 3D, so we can control the dimensionality of the system, and we can also control the geometry. So if we change the angles with which we are interfering those laser beams, we can make for instance something that looks more like a honeycomb lattice or a triangular lattice. So it's great for theorists because we can talk about all sorts of weird geometries of the lattices and connectivities, and then they can try and do it in the lab. And another thing that I know that Professor Javier will be talking about with you is one of the most important experiments in cold atoms in the last decade or so, which was the realization of a superfluid to mottin-sulated transition. And this is the idea that we have such a controllable potential, and as we change the intensity of these laser beams, we can make the potential deeper or less deep. So we can go from having very shallow lattice where the atoms can move around quite easily so they can hop quite easily, and this is a superfluid, to a very, very deep lattice where the atoms are really confined in the minima of the potential, and then they find it so hard to move that they form a mottin-sulator state. So we can do all sorts of things like that. And the other thing that people have been developing very recently in the last few years is about not just making static lattice, but making dynamical lattices. So for instance, this is by interfering different lasers, different frequencies, and we're using piezoelectric. So I'm not gonna talk much more about how they do this, but I want to tell you some of the cool things that we can do with that, because I'm a theorist, so from now on it's going to be a little bit more theory-oriented. And so this is the overview of what I'm gonna talk about. And I have cut a few things out, with respect to the online slides for the interests of time and not killing you after a long morning. So do look online and you will find some more information about these very exciting areas, the SSH model, topological pumping, which I mentioned yesterday and more about probes of topology. And do come and ask me about that too. But what I want to focus on is how we can simulate an artificial magnetic field for cold atoms, an artificial spin orbit coupling for cold atoms, and also give you a flavor for what we're trying to do as a field and where this quantum science is going. So, artificial magnetic fields. Now, as I mentioned, atoms are neutral. So we can no longer just apply the magnetic field and use the fact that that magnetic field couples to the charge of the particles and breaks time reversal symmetry. So as I explained to you in the previous lecture, we have this class of topological phases of matter, which are the quantum whole systems, which have broken time reversal symmetry and these are the churn insulators. And a classic example was the 2D electron gas with the magnetic field. Okay, so we need to be a bit more clever and we need to figure out analogous ways to break time reversal symmetry. And this is really where the field of artificial magnetic fields started with the first idea of rotation. And this is based on a very simple analogy, which is that if you apply a magnetic field or you rotate a system and look at how that system behaves in the rotating frame, you get the same sort of physics. So in particular, if you look at the form of the Coriolis force that you see in a rotating frame, it is a velocity crossed with the rotation. And that looks just like the Lorentz force, which is a velocity crossed with a magnetic field. So the idea is that if you take your Hamiltonian for a rotating particle in the rotating frame, you will have your usual, the kinetic energy. We have a harmonic trap because our cold atoms are usually need to be confined in some way. And then the fact that we've moved into a rotating frame gives us this additional term. And now after a few lines of algebra, you can show that the Coriolis force appears here just like you would expect the magnetic. So the Coriolis force gives you this term that appears just like you would expect the magnetic vector potential to appear in a magnetic field. You know, and you can kind of understand this and okay, a magnetic field makes things go around in a carol way. And when you're rotating something, you're making things go around in a carol way. But one of the main problems of using this approach is the fact that because we have this rotation, we also have a centrifugal effect. So we know it's not just Coriolis force, also centrifugal, and the centrifugal force is giving us this term, which as you can see is reducing the effective harmonic trap. And okay, in some ways this is good because if we can rotate with exactly the same rotation frequency as the harmonic trapping frequency, this term will go away and will be left exactly with the Hamiltonian of particles in a magnetic field. So then we get a perfect analogy in this rotating frame. However, okay, they're also no longer trapped because you've got rid of that confining potential that you are using to keep your atoms together. And so in practice, this sets a kind of a limit. You can try and get around it in different ways, but the best experiment to date has got to very, very, very close to this centrifugal limits. And what happens if you do manage to rotate the gas and see this, then because what you're rotating is a superfluid, and we know that if we put magnetic field on a superconductor or if you rotate a superfluid, you will get vortices. And indeed what they see in the experiments to show that this is really working are these beautiful lattices of the vortices. Sorry, if Omega... Yes, yes, then you just have a trapped... You have a renormalized trap, exactly. And that's okay, but the thing is we want to make Omega R. If we want to get to the strongly correlated regime, we need Omega R, which is proportional to the effective magnetic field to be as large as possible because we really want to be absolutely as fast rotating as we can because the more we rotate, the more effective magnetic flux we have in our system, and therefore we can hope to reach a regime where the number of magnetic flux that we have is comparable to the number of particles, and then we should start to see fractional quantum hole physics with interactions. Unfortunately, even if you do the absolute best that they can do in experiment, you won't get to very low filling fractions. So you won't really be able to see fractional quantum hole physics easily with this. They have tried to come up with some ways around it, but this was one of the reasons that people said after a while, okay, we've done these rotating experiments. Now let's see what else we can do because this is good, but we're not getting anywhere near the fractional quantum hole regime experimentally. So the problem is that if you then have to put in the numbers for what this omega perp is, and if you do, I think you still end up with the difference between, I think it's two orders of magnitude more particles than you have magnetic flux in the system for typical experimental parameters. So a bit of a way. Okay, so the next thing that people thought about was what about using that analogy that I talked about the other day between the very phase and a magnetic flux. So I was mostly talking about this in the context of a Brillouin zone, in which case I was talking about a very curvature in momentum space. So I was making block states that varied over the Brillouin zone, but it doesn't have to be in momentum space. The derivation I did was for any set of parameters. So what if we now really do consider trying to create a very connection and a very curvature in real space, because then they're directly analogous to a magnetic field, magnetic vector potential and a magnetic field. And so this is what people then started trying to do, which was to engineer a real space very curvature. And so how do we try to do that? Well, it means we need to have some eigenstates of our system that we can prepare our system in a particular eigenstate, and then that eigenstate is gonna vary as a function of position. And we want it to vary as a function of position in a non-trivial way, because we know that if we look back at the equations here, that the very connection is telling us, okay, that there is this variation as we move in the eigenstate, and that this is the curl of that quantity. So we need to have some spatially dependent eigenstates. And what people thought to do was to use optical light fields, to use lasers to dress the system. So in particular, all the atom light coupling, you just group together in this dressing Hamiltonian. So we saw examples of this actually in the Ion lecture earlier today, how atom light coupling can work. And now we have just, I've just put all of those possible things in here. I'm talking very generally, and I'm saying there is some atom light coupling. I can ask what are the eigenstates of that atom light coupling? So I just take that two by two or three by three matrix or whatever it is, depending on the number of internal states that the atom light coupling is coupling together. And I diagonalize it to find the eigenstates. And those are now these n of r's. And now what I do is I prepare the system in a given dress state, and then I let it evolve under the full Hamiltonian, which is the dressing, which is the atom light coupling, but also of course the kinetic energy term, which I haven't included just yet. And so it's under this total Hamiltonian. If I have adiabatic evolution, then I will get out the berry connection in exactly the same kind of way that I derived for you the berry connection before. And so this is really kind of a textbook application of that berry idea. And now this berry connection is defined with respect to this basis of the dress states. So what I want is to design an atom light coupling that vary spatially in such a way that when I calculate this quantity in the curl of this quantity, it's non-trivial. And okay, there are some other terms that come in, but I'm not going to talk about that. But this is just exactly analogous to the derivation that I did before. And the thing is that if I were to really go into the details here, I could spend a lecture or two just on this because what atom light coupling you use depends on what atomic species you have. Depends on what levels you want to couple together. Depends on what magnetic field, effective magnetic field you want to come out. And so there's lots and lots of work that's been done in this direction, lots of papers. And so I do refer you to this great review of modern physics by Jean Dalibard and others that came out a few years ago. I just want to mention the experiment that really realized this scheme, which was done in the group of Jan Speelman in Maryland. And again, I'm sorry I'm not deriving for you exactly how this is working, but I just want to give you just the picture in your mind is if you don't have the atom light coupling, so no atom light coupling, then the internal states of the atom are this Mf equals minus one, Mf equals zero, Mf equals plus one of rubidium. So this is the F equals one manifold of rubidium, the lowest internal states. We use this a lot in atomic physics because rubidium is one of the species that we use most. And I think that Professor Javier will tell you a bit more about that. Okay, now what we want is we want to get dressed states. So we need to turn on some couplings. And in this experiment, what they did was to have these Raman couplings. And actually we already talked about Raman couplings earlier today in the Ion lecture, where he told you about how you can, if you detune these Raman couplings sufficiently from the excited state manifold, you just get renormalized coupling between these states here. So basically it's two photon, but don't worry about that. It's just allowing you to change between these states. And now you also have a magnetic gradient in the system, which shifts the different energies of the internal states. And the combination of these two was designed so that if you find the eigenstates of this atom-like coupling, you will get eigenstates that are varying spatially. So this, in particular, its magnetic gradient is varying spatially across the system in such a way as to engineer an artificial magnetic field through a real space, very curvature. Okay, one of the problems with this is that they actually were limited by photon scattering. So you can't unfortunately detune this from the excited state manifold as much as you would like for this species. And so the experiments kind of hit a bit of a brick wall again if they wanted to get more magnetic field coming through. Oh, and this is just to show you that indeed they see the vortices, so it really does work. These are the vortices showing that they have this artificial magnetic field. And there were ways to possibly overcome this limitations, but as these are still theoretical works and I have limited time, I'm not gonna go into more detail, but if you're interested in how to extend these schemes a bit further, then do come and ask me. So that was the status I think by the early 2010s, so about five years ago, and people were going, right, we really need a way to get stronger magnetic fields. How about if we instead go to another limit where instead of trying to simulate continuum physics, so instead of trying to get to land our levels, we try and get to tight binding models like that Hofstadt model that I introduced to you yesterday. And this is then a very different type of approach because now instead of just trying to simulate a Lorentz force or the continuum magnetic field, we instead think how can we address the tight binding hopping amplitudes in a lattice and make them complex. And that's basically what all of these schemes are trying to do. How do we make tight binding amplitudes complex and spatially dependent? Because we know if we can engineer this phase factor, then that's like having a piles phase factor and that's like having charged particles in a magnetic field. And there were at least three, so I'm gonna talk about these three schemes very quickly because these have all been very, very influential in the field and indeed some of them are very influential experimentally as well. And they give you an idea of the diversity of some of the approaches that people are using. So the first two schemes, the shaking and the superlattices plus resonant driving fall into a category of floquet engineering systems. And this is a whole other big field that we're using a little part of but that is becoming very important in cold atoms and other systems more generally, which is saying, okay, we don't really like what we have but what if we shake the system very, very quickly, can we get some emergent physics that is better than what we had to start with? So the idea is that you have some part which is just the static Hamiltonian and then you have periodic driving that you add to this. So this is just saying that it is periodic drive and of course this full Hamiltonian is associated with in general this very quite complicated and involved evolution operator. So we have evolution operator which is time-ordered and depends on the Hamiltonian, the full Hamiltonian that you're applying. But then the idea is what if we modulate the system sufficiently fast that what we can think of doing instead is saying that modulation is so fast we're not really seeing the details of the modulation we're just seeing an effective time-independent Hamiltonian that is being created by that modulation. So it's moving something so fast that actually when you look at it and if you look at it especially every period it looks like it's static. And so this is a huge field and these are some very good reviews about Floquet theory and in general even just at lowest order the simplest thing you can think of doing if you're at very high frequency of the modulation is that you just time average the Hamiltonian that you've applied over a time period and you see what happens. And why we care about it in my field is because you can have the H naught and H effective can be in different topological classes. So if you drive the system in the right way you can end up getting an effective Hamiltonian that describes the system at these stroboscopic time periods that has now interesting topology. And so this is something that people have been trying to do. I also want to mention you actually even get more topological physics. So there are states that come through the driving which actually can be topological and have no analog in the static classification that I described to you the last few days. You can even do more topological physics but I don't have time to talk much more about that. And so the first example that I want to emphasize which is widely used in cold atoms is the idea of shaking. And so this is just a kind of a shaking of the whole system off resonantly with a very high frequency. And so this is again, you can just kind of imagine if you shake system so fast off resonantly that if you look at the system you don't actually see that it's moving because it's moving too fast but the shaking does have the effect of renormalizing some of the parameters in your Hamiltonian. So in particular you can use for instance that time average expression. You can calculate what the shaking does and it will renormalize your J. And it can renormalize your J. I can change the sign of J, the amplitude of J and it can even make J become complex that hopping amplitude become complex. And in particular a very powerful technique has been if you take a hexagonal lattice and you shake it circularly. So this is the same as if you took graphene and you radiated it with circular light field then there is a mapping from this system to the Haldane model which I haven't been able to introduce in these lectures for reasons of time but is described on the online slides. And I just want to show you how powerful this is because this is an experiment which took effectively something like a hexagonal lattice and did this shaking in the circular manner and they were able to map out the whole phase diagram of this model. And this was a model that was introduced by Haldane and he was like, oh no one's ever gonna do this in an experiment because it's very complicated but they were able to see in particular this is as a function of parameters so this is basically a parameter that is related to the fluxes associated with some of the amplitudes here the complex amplitudes and this is an onsite energy difference between two sites. But the important thing to notice is that this red blob is like this phase and this blue blob is like this phase and they were able to see how exactly the topology of this system changed due to this shaking. And so this was a beautiful experiment that was carried out just a few years ago. So we can shake systems, okay. Another thing we can do is we can turn off the hopping amplitudes we don't want and then turn them back on externally. So the idea is is that if you have this optical lattice then you have some hopping amplitudes your atoms are moving between lattice sites but if you then really increase the energy of some of the lattice sites you inhibit the natural tunneling. So you've for instance really increased the sites of these, the energy of these sites and so now you don't have atoms being able to hop from here to here because the energy difference is too high. They're not resonant anymore but if you now turn on some external lasers then you can couple these two sites together. So a bit like with the atom light coupling that we heard before if you have two levels at different energies and you can use lasers to induce the transitions between them. And so that's the basic idea of this kind of scheme. So you turn off the hopping and then you turn it back on. But the way you turn it back on is with lasers and lasers have a phase and that phase because it's a light field the phase is gonna depend on position. So if you do that in the right way you imprint a phase on that hopping. So without the lasers you don't have hopping if you turn on the lasers you do get hopping but now with possibility of spatially dependent phases that can lead to an artificial piles phase and an artificial magnetic field. And this is a very important technique that is used particularly in the groups of Wolfgang Ketteler in MIT and Emmanuel Block in Munich. And I just wanted to highlight that they actually use this to measure in Munich the churn number of the band. So in this way they realized exactly a Hofstadter model so they made hoppings along the y-direction complex and spatially dependent and kept the normal hoppings along x. And they got churn bands from the Hofstadter model and then they were able to see the transport of the system. But it's a little bit different from solid state physics because this is a bosonic cloud. So if you think of fermions all you need to do is set your Fermi level in the band gap and then you have a band insulator the band insulator is in a churn band and you get quantized conductivity. Here we have bosons so what they instead used was that they actually populated the band uniformly by having oh sorry it's cut off there this should be KBT so by having the temperature associated with the atoms being larger than the band width so the temperature allowed you to uniformly populate the band but it was still small compared to the band gap. And just to comment that what they measure then is not a current because in a coal gas we don't have a reservoir normally connected to another reservoir apart from uncertain experiments that you can ask me about if you're interested in but in most experiments we don't what instead we have is this trapped system and so what we can look at are the in situ dynamics of how a cloud is moving and this is slightly different to a current density but can show this churn effect. Okay so then actually a very related scheme to the one I just described and actually one that predates it chronologically was instead of using a super lattice and turning on the hoppings using lasers to use different internal states trapped in state dependent lattices and then turning on the hoppings using lasers so it's the same idea but now we are using different internal states so the details do vary a little bit but the intuition is the same so because we have one internal state here and one internal state here you know that therefore and this is a state dependent lattice this atom can't help here because in this potential, the green potential that it sees this is a maximum so this is not a lattice site for the green and but it is a lattice site for the red so it can only hop now if you use a laser to also change its internal state and if you change its internal state then now it wants to go here because this is its lattice site this is its minimum and so this is again just saying because this is a laser that we're using we have this laser phase the running phase of the laser which imprints a complex phase on these processes okay, I am skipping over a lot of the details but that's just the intuition for what is going on and one of the things that actually has sparked a lot of interest recently and is one of the reasons I talked about this last is because we realized oh, people realized in the field that this is using the internal states to create a synthetic gauge field but there is no real need for these atoms these different internal states to be on different sites so the fact that there is these are separated spatially does not matter here instead we could think about having all of the internal states on the same site and then treating the state index itself as a synthetic dimension so let me try and explain this a little bit more so synthetic dimensions are now after this realization a really general idea in cold atoms and that is that we identify some set of states that are not normally coupled together so these can be the internal states of an atom but they don't need to be and then we couple these together with lasers or some other mechanism in order to simulate this type binding hopping so if we normally look at this then for instance an atom prepared here is gonna stay here but if we turn on this external coupling now our atom can move in this synthetic dimension so we call it a synthetic dimension because it's obviously not a real spatial dimension but in terms of the type binding model that we write down it is like a spatial dimension so we can simulate using these tricks all sorts of different topological physics and the great thing is because this is an external coupling that we're putting in we can choose the phase associated with it we're not constrained to just having the real phases that you would get in a normal unmodulated lattice Yeah, very good question so this is an example so this is how they did it in experiments so how you control the phase depends on which particular scheme you use and this I will explain one scheme so these are the experiments that have been done in Florence and Maryland and recently also in Boulder and the idea is here that for instance this is using the internal states but now the internal states are on the same site in real spatial dimension but we treat the different site indices as if it were a synthetic dimension and so how do we get the phase well because we're using Raman lasers to couple the states together and so one of the things that we heard about earlier today with ions is that Raman lasers are a two photon process and in particular they give you as you have to absorb a photon and that's gonna give you a kick and then you emit a photon and that gives you another kick and so by controlling the angle of these Raman lasers with respect to the lattice as you hop in this direction as you change between this state to this state you have to absorb and emit a photon and that gives you a kick and the kick it gives you depends on where you are in the system and so that is how you get a spatially dependent phase factor so these Raman processes allow you to control the phase associated with these hopping terms because this is an external hopping that we're imposing on the system okay so just to say a couple of other very quick things about synthetic dimension so one thing is it's really cool because this gives us a place that we can finally look at edge state physics and cold atoms so normally cold atoms we have this soft harmonic trap which really screws up looking at edge states but now in this synthetic dimension we have a hard edge and so they were able to see experimentally these skipping orbits that are like the skipping orbits of a classical whole system under a magnetic field they also give you interesting interactions because this isn't a real spatial dimension so you have to think about what the interactions mean with respect to this synthetic dimension and it allows you to go to higher dimensional physics so this is why I bothered to tell you yesterday about the 4D quantum Hall effect because if you have a 3D optical lattice plus one synthetic dimension the tight binding model you write down will be four dimensional and will be as if you had a four dimensional quantum Hall system if you choose the appropriate hopping phases and this is something I work on so do come and ask me if you want to know more I should also say that we very recently realized a dynamical version of the 4D quantum Hall effect using a topological pump which is a little bit different but if you want to know about that then do come and ask me about that I didn't do the experiment but I helped with the theory okay so that was a very quick tour of artificial magnetic fields and cold atoms the other thing that we want to do is get artificial spin orbit coupling so this is now a little bit different before we had P minus oh I forgot a charge here but it doesn't matter P minus A but now what if A the magnetic vector potential is not just a vector field but is itself an ensemble of Pauli matrices for instance now that means that different spins will experience different gauge fields and a simple example of this is you can think about the effect of the magnetic vector potential in momentum space is shifting the minima of a dispersion and now in a spin orbit coupled system simple spin orbit coupled system it means that one spin will be shifted one way and the other spin will be shifted the other way and so this is a classic example of what spin orbit coupling looks like as we draw it in momentum space and so of course you get this for free with electrons this we have spin orbit coupling in crystals and this is an intrinsic quality that they're using in the field of topological insulators to find topological insulators was look for systems with strong spin orbit coupling but now we don't have this because we don't have again a charge so we have to simulate this so what we do is we choose two internal atomic states to act as the spin up and the spin down if you want to spin half system and then we couple the emotional states to the internal states so spin orbit means you couple spin to orbital motion and so that's exactly what we try to do but in this analog way oh yeah so I should mention again I'm not really talking very much about details but there are great reviews that talk about spin orbit coupling and coal gases and just to highlight what's been done experimentally then indeed back in when was it? 2011 in Maryland they realized spin orbit coupling in a particular type of 1D spin orbit coupling using a scheme very very similar to the one I showed you for you simulating a magnetic field again not going through too many of the details but again they're using Raman lasers everything's Raman lasers because it gives us these nice kicks and now the Raman lasers are coupling just two of the hyperfine states of rubidium and what we do is we tune this one out of resonance by applying a strong magnetic field and using a quadratic seamen shift and now we just have these two states and now we couple, we call this one for instance spin down and this one spin up to begin with and then we want to think about this spin being coupled to motion and the Raman laser does that for us because it gives us these kicks and the kicks that you get going this way will be opposite to the kicks you get going this way and that couples the spin and the orbit and so this is just showing you you have to do some tricks to get here but you do indeed get a spin orbit coupling out of this but it's what we call a 1D spin orbit coupling because it only depends on Sigma X if you want a 2D spin orbit coupling it should depend on Sigma X and one of Sigma Y for instance and KX and KY so KX here is just coupling to K naught Sigma X but we would want KY to couple to something like K naught Sigma Y and this is just showing you that indeed they did see the spin orbit coupling and I just want to mention that since then it's also been realized with fermions this was with bosons and very recently it has been done using very complicated schemes for 2D so now we do have 2D spin orbit coupling and cold atoms it looks a bit weird but we do have it and so how is all of this relevant? Well this is just to tell you that we have these tools now we have artificial magnetic fields we have artificial spin orbit coupling and now we're looking at what we can use this for and one of the things people want to do would be to realize a topological insulator with cold atoms and we haven't yet done this we have some of the tools necessary but it's just not been experimentally possible yet so the simplest way that we could do this would be to take two copies of the scheme that we were talking about before for instance creating a churn insulator so and then have one internal state feeling one magnetic field and the other internal state feeling the other magnetic field and so this requires experiments to have very precise control on spin dependent on state dependent potentials because you want to control one internal state in one way and the other internal state in the other way and experiments will get there but it's hard to really get down to the right to cool enough temperatures because you're driving these systems often and so how do you manage to reduce excitations and to cool down with all of these technical details the other thing I want to mention is that people are getting very excited about creating a topological superfluid because we now have the ingredients that are needed in minimal schemes to realize a topological superfluid like with Majorana so I mentioned yesterday this idea of proximity induced topological superconductivity where you had spin orbit coupling and S wave superconductor and the combination of the spin orbit coupling and the S wave superconductor allowed you to get an emergent P wave pairing and that P wave spinless pairing was what was needed for the topological superconductor models that most people are working on. So maybe we have the ingredients but we're not quite there yet and so just as a few perspectives to finish the first part of the talk on, don't worry the second part will be slightly shorter I think. This is what the field is really interested in and we have these ingredients for more exotic phases we want to realize those. There's also other ideas. What about using dissipation to engineer topology? So the fact that we have very controllable systems maybe what we can do is instead of engineering the Hamiltonian engineer the Limblad operators in order to drive the system into a topological phase and you can find more about that in this reference. What about dynamical gauge fields? So this is really everything we've done are really artificial gauge fields because as the atom experiences the gauge field the gauge field doesn't change. As the atom moves the gauge field stays the same whereas if you had a real gauge field it should be dynamical. As the atom moves then the gauge field should react you should be able to have back reaction between these degrees of freedom and people are very excited about trying to put a lot more ingredients in and get to simulating a dynamical gauge field and then maybe even simulate things like lattice QCD in a cold atom setup. That's I think a long-term perspective but that's what people want to do as well. Then this idea of high dimensions that I mentioned and also the real holy grail the strongly correlated topological states. We really want to do fractional quantum whole physics with cold atoms and at the moment we just haven't yet got there we're now stopped by the fact that a lot of our schemes rely on this driving so these different driving processes and that heats the system a lot and so that's very difficult if you want to ground state. Okay so that was cold atoms whistle stop tour now I want to compare and contrast with photons and this is fields that kind of is even younger so we'll be even more bit cutting edge. I don't think anything I'm talking about is older than about well all the experiments the oldest is 2009 most of them were 2013 and so that is also reflected in the fact that there are far fewer reviews so these are the main reviews that we have so far we're actually writing a review at the moment that should come out at the end of the year so keep your eyes open for that and that will be a long review going through all of the topological photonics that we have so far to date. So what about photons? Okay this is quite different to cold atoms. First difference we only have bosons because all photons are bosons so in cold atoms we could do fermionic atoms and then we could think about realizing straight away those fermionic topological states but now we only have bosons so really we're focusing more on those topological states for which statistics doesn't matter. We do have tunable effective interactions to a first approximation photons are actually non-interacting I mean we see that we don't see photons interacting around us if we put them through a non-linear material then we can get some mean field interactions if we couple them to level atoms or emitters or we can get strong effect of photon-photon interactions so this is what the field is really wanting to do more of in the future but everything I'm gonna talk about today is non-interacting photonics. We can also design structures for light so we're talking about like moving through materials we can really design those we're very good at that. Another main difference is that most of the structures I'm gonna be talking about are inherently driven dissipative systems so they're really out of equilibrium physics. So cold atoms is really an isolated system to a good first approximation. You can kind of neglect atom losses but now we really can't neglect atom losses and these losses are a great tool for us because we're constantly losing photons from our system and that means we can collect those photons and look at what's happening and you know this because as you shine light then the light is being scattered and is leaving the structure and if you look at the scattered light then you gain information about what's inside. So in particular we have what we call the far field imaging schemes which look at light far away from a sample and that's a bit like time of flight in cold atoms so because the lights had to travel to get to you you are sensitive to the momentum of the photons or you can put your detector really close to the sample and then the light isn't traveling and so you're just seeing where the photons are and so this gives you access to the real space wave function of your system. Another thing before I jump into it I want to highlight a few important things about this field is it's really hard to talk about one set of systems because photonics is a big field and it covers quite a large part of the spectrum so in particular if we're talking about topological photonics we have everything from the microwave regime where you're talking about structures designed on a centimeter scale so that's like you look at it and it's centimeter spacing between the elements of your photonic crystal right the way down to the infrared and the visible regime where we're thinking more about for instance this is a picture of silicon photonics platform so this is etched into silicon and the length scale here is like 10 micrometers so there's really big differences between the types of structures that all come into this heading of topological photonics and I also want to apologize that I don't have time to talk about everything so metamaterials and polaritons are also important in this field but I'm gonna leave those because for interest of time another thing I want to emphasize is this is the point where I fess up and I say well I'm not sure I actually should be in a quantum science workshop because topological effects aren't actually all quantum and in particular the ones I'm talking about today are for instance associated with churn insulators or churn bands are not intrinsically quantum and what I mean by that is I was talking about the topological classification of these kind of matrices so that's what we went through but actually this matrix can be any matrix that describes normal modes in a system and this means that there's been a big explosion in the last couple of years really the last couple of years talking about topological band structures in anything that has normal modes so classical photonics which is really what I'm talking about today phononics and mechanics and so this is a picture of an array of pendula and the coupling between the pendula engineers band structure for the phonons in the system which has got topological properties so you can see robust edge states here so this is a view that talks about some of these other things and just to say it's not actually quantum some aspects of it will be and if you want to add interactions and do fractional quantum Hall or use Fermi statistics that is quantum but if you just want churn bands then we're gonna do that with classical systems okay so this is what I'm gonna talk about in the half an hour or so that I have left and in particular the quantum Hall systems of photons quantum spin Hall systems of photons and then some future perspectives and again this is the stuff that I cut out but that you can find on the online slides and do come and ask me if you want to know more okay quantum Hall so this started, this field started with the proposal by Haldane and Raghu of how you could engineer a topological photonics system and they were really the first people to point out that you just needed normal modes you could just do this with photonic energy bands and so how to see what these kind of photonic energy bands are this is just a very basic just meant to very basically show you where we get that normal mode equation coming through with electromagnetism if you take Maxwell's equations without the source terms and then you assume that you've just got linear isotropic loss free media just simple you have the permittivity and the permeability, tensors entering here and then you can find the normal modes of this system and this is just taking the inverse of this over to the other side and you can see this part here is now like that H that I wrote before so what Haldane and Raghu said is okay we have a system with normal modes that is electromagnetism how do we make this thing topological and their idea was to say let's think about what we do in condensed matter and try and do the same thing with photons so what do we have in condensed matter for a quantum Hall system we have electrons in a periodic potential okay so let's take photons in a periodic crystal so photons feel the permittivity and permeability the dielectric properties of a material so if you design the dielectric properties in a periodic way they will feel that as a potential the other thing they said is we want a magnetic field we want to break time reversal symmetry photons are not charged and so how do we well okay fine photons are also not really the same sort of particles as electrons but how can we break time reversal symmetry well we can do it using certain types of materials that break time reversal symmetry and these are called magneto optical materials all that really means is that if I apply an external magnetic field that the optical properties of the material are affected so the dielectric tensor and the permeability tensor really change depending on this external magnetic field and because as I said before the external magnetic field is a great way to break time reversal symmetry the fact that there is a coupling with that external magnetic field and the dielectric properties of this medium breaks time reversal symmetry in this medium and that's what they did very soon after in MIT experiments where they took a lattice so this is the periodic part a lattice of ferrite rods and these ferrite rods have the property that they are gyromagnetic when you apply a magnetic field so they applied a strong perpendicular magnetic field and then they broke time reversal symmetry so now if you inject light into the system the system sees this periodic structure but one that is not time reversal symmetric and I'm afraid I can't derive it for you because they actually did the optimization of the structure and everything numerically but they numerically figured out what kind of lattice would give them bands so photonic energy bands that had non-trivial topology and then they saw okay if we make this structure we get these energy bands we have an edge state and this was their proof that that edge state was a topological robust edge state so what they did was they pumped the system and then they looked at what radiation was transmitted through the system and you can see here that the radiation if you pump it here is only going in one direction so that's like a chiral edge state indeed it's pumping resonant with this edge state at the edge and if they put a defect so they put here this defect that should backscatter so a photon going to that should normally backscatter but because this is a topological edge state there's only one edge state so there's nowhere for it to backscatter there's no backwards traveling mode it has to go all the way around and continue and so this was the first example of a topological effect in photonics and I also just want to mention that these experiments have advanced so quickly that now we can even do things before condensed matter so photonic crystals based on that were actually done by this group but now generalized up to three dimensions not time reversal symmetry breaking were the first to observe vial points which you may have heard a lot about and before condensed matter condensed matter did it just afterwards but we can now do things before condensed matter okay so that's the end of the story right I mean we apply a magnetic field we have a gyromagnetic system it breaks time reversal symmetry we have churned bands for photons not so fast so the problem is that the effect that this is using this gyromagnetic effect is very very weak at optical frequencies so that was using microwaves the structure was on centimeters if we want to do the same thing for optical frequencies so this part of the spectrum we can't and also if we want to do this in a silicon photonics setup where we want it to be used for a practical photonics device then we don't really want really strong magnetic fields there if you want to use silicon photonics and put this on a chip that you then have in your computer you want to keep it as simple as possible and so we turn to other approaches for breaking time reversal symmetry that are now very similar to the cold atoms ones so now you can probably understand why I'm standing here and talking about both because the underlying idea between the two fields is actually very similar even though the actual physical system looks really different and so floquet engineering was one of the first successful approaches to realizing this time reversal symmetry breaking at optical frequencies and this is just very quickly to say this is a particular type of system that's called a propagating waveguide array so what this means is that we have basically waveguides for the lights the lights propagating along the waveguides and we make certain approximations to Maxwell's equations namely that it's mostly propagating in the z direction so it's not really propagating in other directions it's mostly got the wave vector along z and that it's got a slowly varying envelope so it's not moving very it's not varying very quickly compared to the structures of the system and if you make these assumptions and you do a bit of math then what you find is that the evolution of light in a propagating waveguide so just from Maxwell's equations is actually the same as the Schrodinger equation but with the role of z and t reversed so where you would normally have i d t d by d t of psi you now have i d by d z of psi and that comes from this these curl things up here and a few vector identities so now where you would be modulating time what you can do is modulate something in space and this was realized in the group in this group in the technion in Israel where they took a lattice of propagating waveguides and they shook the waveguides by modulating the waveguides changing the refractive index of the waveguides along the propagation direction and so by this analogy this z t shaking so shaking the waveguides as they travel moving the waveguides as they travel is the same as shaking them in time and therefore realize that Palday model that I was talking about in Cold Atom so you see it's really similar fields here and they saw indeed the presence of Karel edge states oh I shouldn't say they can also do more again so we now do anomalous floquet topological states with these systems this is a particular type of system but one of the other things we can do is taking ideas from the synthetic dimensions that we mentioned before so this is just the same slide because it's exactly the same idea and actually it was people transferring ideas between fields so we thought of synthetic well not me personally people sort of synthetic dimensions in Cold Atoms and then we came along and said oh well why don't we do that with photons so we also can have sets of states that we can couple together and there have not been any experiments doing a synthetic dimension for photons yet but there are lots of ideas so one idea would be to use optical mechanical systems another idea would be to use optical cavities just to explain this very quickly this is the idea that we were very fond of which was to use the different modes of a silicon ring resonator okay so a ring resonator is basically a circular waveguide for light and it's a resonator because if the light has an integer number of wavelengths fitting around this ring then that's resonance condition okay so it means as a function of frequency you will get peaks of resonance peaks so there will be certain frequencies at which it doesn't resonate and then there will be other frequencies at which it does resonate and as you increase the frequency you will see successive peaks because it will fit more wavelengths around this ring and this defines a set of modes so these are the different resonance peaks of a ring resonator and so normally these are not coupled so if you shine in light at this particular frequency with this angular momentum then it will just go round and round and it will stay in that angular momentum mode that resonance mode oh this is just saying that this is the frequency comb don't worry so much about the equations but then our idea was to couple those states together using basically an external modulation so kind of a floquet engineering type idea where we put some external driving into the system that allowed photons to hop between modes and because we were doing this externally because we were having to add something to the system we had the opportunity to add a complex phase factor so coming back in this example we were able to by controlling the angle with which we shone the pumping field on our system that allowed you to do this then we were able to implement spatially dependent complex phases so that's just an idea to say yes we can do synthetic dimensions with photons and this means we can also say what about 4D topology in photonics and who knows who knows whether cold atoms of photonics is gonna win at the moment we're seeing which horse is gonna come in first the other thing that we can do which I really like about photonics is we can think about practical applications and so I just wanna say a bit more about this on this slide which is that now we have real systems that can be all sorts of frequency ranges or all sorts of length scales and we have chiral edge states and chiral topological edge states are so these are the edge states associated with a quantum Hall state of light they're going in one direction they can't backscatter they're localized at the edge of the system okay well that's a great way of guiding light where you want light to go so if you want light to turn around that corner and to go over there then you just design the edge of your system to give you the corner and to direct it in another way and this means that now as you put in disorder or if you put in fabrication imperfections the light doesn't come back at you it keeps going it has to go around the defects because these are robust topological states because of the topology of the photonic bands and so we're hoping that maybe this is a way of making more robust optical devices so a device in which you send light in one direction and it won't come back at you and that's really useful because we have certain applications like optical isolators where you want to protect for instance a laser you want a laser to send out light but you don't want the light that the laser sends out to come back at it because it's too delicate and that light can harm the laser and on the laboratory bench you don't need to worry about that at this length scale you know you can have an optical isolator that's like that big and you're fine but if you want an optical isolator that is on the micrometer scale and that can be on a silicon chip we don't know how to do that yet and this topological edge states is one of the ways one of the reasons that we like it so much so it's actually got applications that are quite interesting another thing just to say very quickly is that now we're really not like solid state systems because of this driving and dissipation so in solid state systems to see the whole conductance we filled a band using the Fermi energy and then we looked at the conductance now we have driving and dissipation and so how do we see the topology the topological invariant of the energy band in a driven dissipative system and so there's been at least a proposal by some of my collaborators which was to talk about the displacement of the photonic steady state so this is using the losses that we have in order to excite many states within this energy range so you pump at this frequency you use the losses to excite a few states and then the steady state should represent the eigen states in this band in some way talk to me if you want more details because that's just a quick flash okay so now I'm doing quite well right I think I should be done by 130 don't worry lunch will come but let's first talk about quantum spin-haul systems of photons so this is another name as I said for topological insulators right so I told you yesterday that topological insulators quantum spin-haul systems where we have two edge states on an edge with one going in one direction and the other going in the other direction is protected by time reversal symmetry for fermions the reason that we had these protection of these edge states was because we had Kramer's degenerate Kramer's theorem which said that these edge states have to be doubly degenerate and at this point this is a robust crossing of these edge states that can't be lifted because of Kramer's theorem but I also told you that for bosons T squared is not minus one it's plus one and that means we don't have Kramer's theorem and so we don't have topological protection if we make the system with photonics it is not robust anymore but what we can do is we can try to make a system that basically suppresses any process that would couple these states and that's so it's not real protection it's just saying well if we can get rid of all of the processes that would allow a photon to go from one, whatever one spin state means in this case spin can be like polarization or something like that or direction to another then we can kind of say that we have this system and so there have been some very important experiments that fit into this kind of class that aren't really topologically protected but are inspired by topological phases of matter and the most important experiments or one of the most important experiments in 2013 was by Mohammed Effetsi and others in Maryland and what they did was take this silicon ring resonator array so in these site resonators we have this resonance condition going on that I talked about so the light that is resonant with the site resonator spends a long time in there and then these link resonators are designed to be off resonant so light can travel through the link resonators but it does so very fast it doesn't have time to it's not resonating so it doesn't go all the way around it just hops in and hops out and this can actually be described to a very good approximation with a type binding model where we ignore these link resonators and they just mediate a type binding hopping which is another reason why theorists love photonics and the idea that they had was to control the position of these link resonators and to displace them such that if you have photons so the photons are spending a lot of time in these site resonators and then going quickly around these links that a photon going in this way around the system in this direction so counterclockwise travels a different path a different length then photons going in the clockwise direction and the fact that the photons travel a different distance means because photon so light has a phase e to the i k dot r if it has a different distance that it travels it gets a different phase okay so r is different between these two paths which means there is a relative phase difference this process compared to that process and that's the effect of a magnetic field actually but recast in this language so you can see that from the piles phases so the piles phases are also a different phase going one way to the other way because of the magnetic flux so they made this distance displacement vary so as to engineer a magnetic field and this is like a churn insulator in that we're applying a magnetic field but the thing is is that you have modes that circulate one way around these site resonators but you also have the modes that can circulate the other way around these resonators so what I mean by that is a mode can go like that but there's another degenerate mode that can go like that and that provides a kind of two pseudo spin degrees of freedom and so for modes that have for instance going like that in the sites they had a magnetic field that say pointed up but for the mode that was going the other way they had a magnetic field that pointed down and this is necessary because they haven't broken time reversal symmetry here if you look at the whole system it's time reversal invariant so what they did was they just pumped one mode and they tried to reduce all of the scattering that could couple to the other mode and this worked quite well so this is just showing you that indeed they saw edge states so they pumped here and then they saw the light coming out here and this is experiment versus the simulation and even if they added a defect they knocked out one of those resonators they saw the light propagating in one direction because they were only pumping they were only exciting this mode and they neglected that mode so it's not topologically protected but it kind of is topologically inspired another thing just to finally mention is an idea which is like the rotation and this is a beautiful experiment done by John Simon's group in Chicago where they take a multi-mode optical cavity and then they twist the cavity so instead of just normally a cavity is a mirror here and a mirror here and light bounces back and forth now they add a couple of mirrors and they twisted out of plane such that as light travels in this cavity it actually rotates so if you think about a particular orientation of the lights and then you take that image and you send it around this cavity then as it travels it undergoes a rotation because of the way they've twisted the mirrors out of plane and actually if you do the description of this if you do the mathematics this is analogous to taking gas mass of particles and rotating it so it's like rotating a coal gas and what they were able to see is land our levels of photons so this is like a continuum scheme there's no lattice here and these are actually all the different angular momentum states that in a land our level are all degenerate and they were able to tune these angular momentum states to degeneracy and to build up in that way land our level so this is just as a function of some residual harmonic trapping but the thing to really emphasize here is that it's the same idea as that rotation but now for light and it's kind of a spin hall because this rotation is one direction for light traveling through the cavity in one sense but it's the other direction for light traveling in the cavity in the other sense so there are two modes here but you just excite one of them and if you only excite one and there's no back scattering then you will see this magnetic effect and this is really promising for strongly correlated photonics because the same group is currently trying to do a Rydberg EIT scheme which is basically taking a glass of Rydberg atoms and using their strong interactions to create effective photon-photon interactions so they want to put a load of Rydberg atoms in this and then they think they're gonna get a fractional quantum whole state few particles but a fractional quantum whole state of photons and John I think is saying that this should happen in the next couple of years so watch this space we might soon have fractional quantum hall in photonics and beat the cold atom people perspectives so this is just the final slide then to say okay where are we standing at the moment with photons so photons as I've tried to explain to you have got wide variety of different setups different frequencies different physical mechanisms but generally speaking we've been able to do quantum hall effect systems very well at the moment but we'd like to really bring this to the point of making practical devices so moving towards I would say maybe not quantum technologies but quantum inspired technologies because we're using the quantum physics insight to make a classical device we'd also like to think about this driving in dissipation so can we engineer the topology through dissipation because we have non-equilibrium systems we also have these bosons I didn't mention here any analog of a topological superconductor because okay superconductors are inherently associated with fermions so there's some debate in the field at the moment about what we can try to do with those kind of symmetry classes and the big question that everyone is also talking about is again this fractional quantum Hall so we want the fractional quantum Hall because it's really cool it's strongly correlated and also might have interesting interactions and the idea would be for instance to take something like the setup of Mohammed Fetzi with these silicon ring resonators and add some two level atoms to it or to take the setup of John Simon with this twisted ring resonator and add some Rydberg atoms and then see how to prepare a fractional quantum Hall state now there are also interesting questions here because how do you prepare a ground state with photons? We don't have a chemical potential so this is something that people have been working at how to do driving schemes that could try and drive you towards interesting strongly correlated quantum states and so we do want to become quantum with photons so we do want to put the quantum back in and that's one of the directions that this field is going and I should just mention very briefly people are also asking can this be useful for entangled photon transport if you give us some entangled photons what happens to those in these edge states and the answer is maybe it's useful but we don't really know yet okay and so that brings me right on time to the end of what I've tried to tell you about in these lectures so I gave you quite a long introduction to some of the ideas of topological phases matter and I hope I've also shown you that even though this is a big buzzword and condensed matter that we should also be paying attention to it in cold atoms and photonic systems because we have the opportunity to engineer and really control new types of systems we've got new observables and especially with photonics we have the prospect of making practical devices based on this and we hope very soon to get to the strongly correlated quantum regime where we can start talking about fractional quantum Hall states of photons or atoms thank you very much