 It might be that we start a bit late again. For the morning session is Andrea Armour from the University of Nottingham. And the title is Multifotone and Multimode, Josephson Photonics. So please, Andrea, you have the floor. Thank you very much, Chipman. And I'd like to start by thanking you and all of the organizers for inviting me today and for organizing such a fantastic meeting with so many really stimulating talks this week. So I want to talk about some theoretical work on exciting photons and cavity modes using Josephson Junction circuits in cases where multiple photons or multiple modes are involved. And this was work done with students Kiran Wood, Grace Morley, but the main player was Ben Lang, my postdoc who's with you in the room and who spoke on Wednesday. So do grab him at the coffee break if you want to hear more. So I want to start really by taking you back in time to Wednesday morning, when we heard some really nice talks on this system, a voltage biased Josephson Junction coupled either to a microwave cavity supporting multiple modes or just a single lumped element oscillator. And what we heard was that the voltage bias provides an energy which the Cooper pairs can dump into modes of the cavity. And so we can get inelastic tunneling of the Cooper pairs giving rise to DC current, but we can also have excitation of the cavity modes. And there's a very simple model that can be used to describe the system. If we have several modes, we can just have them as harmonic oscillators. And then the Josephson Junction part, we have a Josephson frequency set by the voltage and then essentially the displacement of all of the oscillators with a zero point fluctuation strength here which is basically given by the set by the impedance of the mode, the particular mode. And of course there are also losses that we have to worry about in this system as well. But one way of looking at it is to think of this as a nonlinear oscillator system. And in that sense, the Josephson energy acts as a control parameter. It controls the strength of this drive which because of the presence of this term is strongly nonlinear. And so we can think about what happens as we change that parameter as well as some of the other parameters in the system. And of course I'm gonna focus on this particular system but there are lots of other systems with Josephson junctions in squid geometries which can be addressed with different types of biases which would give rise to similar behavior. So this strength of the quantum fluctuations of a particular mode is a really crucial parameter in the system. If it's very small, then these systems of course are nonlinear but the nonlinearity will become apparent at large photon numbers. And this suggests that these systems will behave quite semi-classically. That's not to say that they won't have important quantum properties but the classical dynamics will be important. And then we can think about maybe the quantum fluctuations on top of that. If Delta naught is of order one, and this is something which has been achieved in circuits, then that nonlinearity will become apparent at a level of a single photon or less. And so really there won't be a regime where these systems behave nearly. There'll be immediately strongly quantum in their behavior. And I want to focus on the dynamical transitions that these systems can give rise to, down-conversion time-translational symmetry breaking. And I want to focus on two particular cases. Firstly, a single mode case with a very large quantum fluctuation strength, like this device that was developed in the satellite group and Denny talked about on Wednesday, which they've used to carry out a series of very beautiful experiments and in particular looking at multi-photon resonances. And so that's the sort of system I'll have in mind talking about multi-photon resonances. But in other systems, straightforward cavities where there's been no attempt to engineer a very large inductance, this parameter is very small, typically. But these can still be interesting. And in this case, there was some very nice progress from Alex Rimburg's group who developed a way of basically inserting a voltage bias into a cavity with some Cooper pair, some Josephson junction device embedded within it in a way that preserved a really high Q. And so these systems are also interesting, but in this case, we have to worry about not just one, but multiple modes in the physics. And I'll talk about those towards the end, depending on how much time is left. So starting with a single mode model, we can think about tuning the voltage and what's going to happen. And so as we tune the voltage or equivalently the Josephson frequency, we will be able to create photons in that mode whenever that frequency matches an integer number of those photons. And so we expect to have particular resonances. And of course, these were seen in the Sackley group and indeed elsewhere. So close to those resonances, we can simplify the Hamiltonian. We can get away from the time dependence by transforming to a rotating frame. So basically the time dependence then sits on the operators here, they start to oscillate. And so we have an oscillation within an oscillation. And so it's perhaps not surprising that there's a way to write this in terms of Bessel functions. And that's this form here, which should be valid close to a resonance where P photons are created. And so we have creation operator for P photons and annihilation operation for P photons. And then this Bessel function part, which looks a bit complicated, but really that's just telling us that once the cavity mode is excited and there are photons in it, that will act to affect those creation processes. And there's also normal ordering here, which just tells us that we don't really need to worry about the ordering of the operators in here because it's been predefined with the a daggers or on the left. And writing it this way with the Bessel functions is actually very helpful if we want to go into a classical description because we can immediately write this as a classical Hamiltonian with amplitudes instead of operators and Bessel functions here. And of course, all of the classical behavior is very strongly rooted in Bessel functions. We can detune it a little bit if we like. Ordering the operators at the beginning gives us this slightly renormalized josin energy depending on the quantum fluctuations. The other thing to note about this Hamiltonian is that once we're in the rotating frame and we've also dropped terms that aren't resonant, making a rotating wave approximation, we have a Hamiltonian which now has a rotational symmetry and we'll see that is important later on. We can take that Hamiltonian and combine it with descriptive terms to write down a master equation for the system. And that gives us a theoretical description of the quantum dynamics. So why bother with going up in P with this system? So processes where single photons are created or two photons are created, these occur across a wide area of physics and quantum optics textbooks are full of these things. But they don't talk about processes where six photons are created. Those are really quite hard to find. And it's really exciting, as we heard on Wednesday, that you can actually get this to work and generate these packets of six photons up to six photons at a time as the Sackler group did in collaboration with the Olm group and published in this recent paper. So, but what's interesting perhaps beyond the fact that it's just a really exciting thing to be doing. It's a novel thing to be doing. What might we get from theory that would point to interesting features going up in P? So the three photon resonance P equals three in this and similar systems actually been explored quite widely recently by different groups, in particular the Olm group. And we also heard from Lisa on Wednesday about this as well and her analysis of this very nice description of what happens in the three photon case. What I want to talk about is what happens when you go beyond three. And our claim is that qualitatively new features seem to emerge in the quantum dynamics that make this really worth looking at. So let's start with a simpler case, a two photon case. So this is perhaps not very far from the kind of things that are common in quantum optics. And let's look at, so this is the Hamiltonian in this case, the P equals two case. And let's look at the average number of photons in the steady state of the system as the Josephson energy is increased. And the first thing we can do is carry out a classical analysis. We can turn these operators into amplitudes and solve classical equations of motion. And that's what this thin gray line is here. This tells us that we have a threshold and the system will get excited. And then there'll be another bifurcation and we expect the amplitude to settle down. So that's what the classical system does. It gets excited and that excitation is at a very low photon number. Typically it doesn't really need the Bessel function. And so it's this fairly well-known parametric process. We can go beyond that and do something which is semi-classical. We can take the Hamiltonian, the master equation. We can transform it into a phase space. Let's look at the dynamics of the Wigner function but not take all the terms into account. We can, the nonlinear dynamics and the quantum aspects give rise to higher order derivatives when we do that. But if we drop those and we just describe the system using a Fokker Planck equation, it gives us something which essentially has semi-classical dynamics but also zero point fluctuations and noise. And that's what this dashed line does for us here. Finally, we can solve the full master equation and we see that agrees pretty well but not exactly with the semi-classical analysis in this case. And in particular, that initial sharp classical threshold is smeared out but it does settle down eventually. And the way the system saturates in the end depends very much on this, the strength of the quantum fluctuations. So if those are weak, then this end value becomes very large. So when you make the quantum fluctuations large, you shrink the phase space of the system. Everything happens at smaller photon numbers. And that's very important for understanding behavior. So that's the two photon case. What about going beyond that? So we heard from Lisa about the three photon case. Here, when you look at the classical analysis, there's no smooth increase. There's a bifurcation and a fixed point appears at a finite amplitude and then it locks in as well. The origin remains stable but its basin of attraction gets really, really small. And so again, you get fairly good agreement here when you compare the fully quantum and the semi-classical. The other thing to notice is that the photon number has increased. So as you increase P, the phase space grows naturally. So the level at which the system settles down gets bigger at higher P. When we go to P equals four, things are really starting to change. And this suggests that there's something interesting that's happening at larger photon numbers. So now there's considerable disagreement between the semi-classical and the quantum case. And indeed, it's not even monotonic. The photon number goes up and then for a little while, it goes down again before settling again. And again, as we're increasing the photon, the order of the process, the level at which this is happening is going to higher photon numbers. So this is the sixth photon case and I just wanted to show you some of the features of this. So this is showing in color here, this is the number distribution. So this is N along here and the color is the probability in the steady state for finding that number of photons in the system as we change Ej. And again, we've got some classical dynamics as a guide here and then we've got the average occupation. And what you see is that as Ej is increased, the system starts at the origin but then it develops some probability to be away from the origin but then this falses for a while and the system seems to go back towards the origin before it settles down again, apparently at a higher amplitude. And we see as this is an oscillation in the average occupation number. So it first increases around the classical bifurcation. It has a second peak later and then it settles down. We can look at this in the phase space and these diagrams are a little bit busy. They've got classical flow lines and fixed points but the blue color is just the Vigna function showing what the system is doing in phase space as we change Ej. And essentially as we start off down here, the system is around the origin. And then because of the P-fold rotational symmetry in the Hamiltonian, as we go into the bifurcation, the system is spreading out with these six bright points around here but then as we continue, it goes back again. So we have this oscillation. So this bifurcation, this dynamical transition is actually very complicated that we're seeing here. If we go a bit further, in fact, there's a second bifurcation where these six points would actually split into 12 but we can't see that just yet on this diagram. So I think the message I want to convey about this and why I think it's really interesting is that if we look at what happens as we change the quantum fluctuations and we essentially tune the phase space of this system, we don't just increase this effect by making the phase space smaller. So this is a scaled version of the occupation number as a function of Ej on a log scale with different values of the quantum fluctuations. And what we see is that at smaller values, 0.4, there are no oscillations visible, there's nothing happening. As we increase it, we get a series of very sharp oscillations and this is shrinking the phase space but if you keep shrinking it, they start to wash out again. They're not quite as clear at these largest values. So what I think is really interesting about these oscillations is that they don't occur just in a strongly quantum few-photon limit. Of course, they're washed out in a limit with many, many photons but they somehow they exist in some mesoscopic regime in between where the photon number is neither too large nor too small. Of course that makes it very tricky to analyze it as well. So let's go beyond the static features and talk about some of the dynamics going on in the system. So it's described by a master equation and the right hand side of that is captured by a L'Ovillian that acts on the density operator and the timescales of the system, its dynamical timescales are all within that L'Ovillian and if we calculate the eigenvalues then we know all about the timescales of the system. Now, what we expect is as that we go towards something that in the classical cases of bifurcation we may expect to see a slow timescale emerging, a signature of critical slowing down essentially. Now, people in the nonlinear dynamics community think about large photon limit where the number of photons is very large and in that case you can start to have a timescale that doesn't just become very small it tends towards zero in that limit and people think of that as then becoming in some sense a phase transition. So let me start with an example and this is one of the kind of models that David was solving yesterday. This is a two photon process and in this case, as you increase the effective drive the system goes from sitting around the origin to suddenly developing a couple of nodes at the side. So they appear at a finite amplitude making it a first order transition and it breaks the symmetry and eventually as you keep going with the driving they disappear. And if you look at the timescales in the system as you go through this bifurcation you get a couple of timescales becoming very slow and then one of them goes back up and the other stays very slow and that's really a signature of the fact that you essentially have these two lobes and there's a very slow timescale associated with switching between them. So this is a sort of model that people discuss and it's a relatively simple one with a two photon process. What about going up in photon number? What happens in those cases? Well, for p equals three, so n here is the photon number and then we've got the timescales here on a log scale becoming very slow here and we've got a timescale that comes down and goes up again around the point that the system gets excited around the bifurcation and we also have a couple of timescales that go down and they stay down. They wiggle a little bit but essentially this fits this paradigm that people have discussed very widely of dynamical transitions and we would expect that if we could make the phase space bigger then this would come down further and this would go down further and the thing would look sharper and tend towards something a bit more like a phase transition. But what if we increase p? Well, if we go all the way up to p equals six it gets much more complicated. Of course, if we're looking at the behavior of the occupation number this has got these oscillations. And what we see is that we've got several timescales that become very slow but instead of going down and staying down and one coming back up these timescales stick together and they have these oscillations and these oscillations match essentially perfectly what's happening in the photon number. And in fact, they were present back in p equals three even though we couldn't see any signature in the photon number. So really this is telling us that the dynamics is leading to this oscillation and there are signatures in these timescales which are much more complicated the kinds of dynamical transitions seen in simpler oscillators. Now, of course, once we have slow timescales we can simplify the problem and we can start to ask what these eigenvalues mean physically and I don't want to go into the details but what Ben found was that essentially you can match up the oscillations in the number with oscillations in a timescale which basically describes the rate at which the system goes from one of these points at a large amplitude back towards the origin. But it's a difficult system to describe and we don't have a fully intuitive description of why these oscillations have arisen and that's something we're still working on. But let me now talk just a little bit about the multimode case and this was stimulated by some experiments that were in the Cohen-Hurven group and in this case they took their device and they embedded it in a high-cutivity using the Riemberg approach to bias the voltage without damaging the queue. So they had the DC voltage bias here and they looked at the emission from the fundamental mode as a function of the voltage and they did this then sweeping up to levels where the voltage corresponds not just to where you would expect the fundamental mode to be but of order 10 times that. So this is an enormously high voltage compared to the fundamental frequency and you might think, well, could this be some 10-photon process? But absolutely not because the quantum fluctuations here are very, very weak and the multi-photon processes depend on powers of the quantum fluctuations. So they're very strongly suppressed. But what the system does have is lots of modes and these have a high queue. And so in this case, there's something going on with the mode. So basically one mode is becoming excited and somehow the energy is getting transferred down to the fundamental mode where we're seeing this strong emission coming from. And so that raises lots of questions about what happens in these multimode systems. And we go back to that initial Hamiltonian but now we have to worry about the sum here and the different modes involved. And when we look at this at first sight, we see that the point is that the Josephson junction here couples together all of these modes. And so naively, you might think, well, this is some kind of all-to-all coupling. They're all coupled together. So maybe some kind of mean field theory might be appropriate here. But this isn't the case at all because these modes are all oscillating at particular frequencies, which are different. They also have different zero point fluctuation strengths of the way that they get coupled together depends very much on the frequencies and the Josephson frequency itself. So most of the coupling won't be important and it isn't a mean field problem. So to address this, we can think about a simple model in which we excite at some harmonic of the fundamental mode and we have an ideal dispersion with each of the modes at integer values of the fundamental. We assume a constant damping and we assume that there's some number of modes that's relevant. Of course, there will be some cutoff in the system but that will be smooth but just to make life easier, we're going to assume that that's a hard cutoff. And this was the kind of model that Ben was talking about on Wednesday but I'm going to look at it from a different point of view in terms of where this transition comes from, how the fundamental mode gets exciting. And so the place to start with something that's complicated is the classical dynamics. So turning these operators into amplitudes and we've then got this Hamiltonian and exploring what happens numerically essentially. And one neat thing to look at is the total phase across all of the oscillators. And as you increase the Jowson energy, the system goes from oscillating at the drive frequency set by, with a period set by the Jowson frequency to the period set by the fundamental mode. So what you get below the threshold is this red curve which has sort of sawtooth oscillations with period set by the Jowson frequency and then that period shifts. So the period in the Hamiltonian no longer describes the oscillations above the transition. And this was something that was looked at by Simon Cooper who actually found that despite the fact that this is a very complicated problem, you can find an analytic description for this state, this sawtooth state that emerges with the period of the fundamental mode if you make some approximations. What they didn't look at was exactly how this transition occurs and when it occurs. So if we look at this classical dynamics in systems where we now we excite here at a frequency that matches the third harmonic. And we have a system here with 11 modes and another one here with 12 modes. And then we ask what happens basically as we increase the Jowson energy. So initially we have the mode that's resonant with the Jowson frequency excited but also the mode that's at twice that frequency and the one that's at three times that frequency also gets excited as well. And then we have an abrupt transition where all the modes get excited including the fundamental. And in this case it's continuous transition with the amplitude of the fundamental mode going continuously. But if we change the number of modes then it's no longer a continuous transition. It becomes discontinuous, there's a sudden jump and what you see is that in fact it's suppressed. This transition occurs quite a lot later in this case. Now, if we want to try and analyze when these occur this discontinuous transition is very difficult because we don't really know where it's going to settle down to. This case where the transition is continuous is much easier because all we have to do is find out where this state where most of the modes are not oscillating becomes unstable. And so that's something that we can seek to address. What we do need to know is what the amplitudes are what fixed points are for these modes that get excited early on. So there's some small subset of the modes that will get excited without a threshold. And we need to know what their amplitudes are but we know that the other modes start from zero. And so that makes locating this transition feasible. And so that's something that we've worked on a bit. So these continuous transitions when they can occur occur at relatively low EJ compared to the discontinuous because they boil down to relatively low order processes. And those are favored when the quantum fluctuations are weak. We can analyze this by thinking about a certain set of modes which are excited the resonant mode and the harmonics of that and modes that aren't excited at all. And we can again use the rotating wave approximation. And what we see is that the modes that are not excited initially fall into different subspaces. So for example, here we've got 13 modes and we're exciting in resonance with the seventh one. And there's a subspace involving one, two, three, four of the modes. And that subspace is where the excitation starts at the transition in this case. So in fact, not all of the modes necessarily go at once at these transitions, they split into subspaces and if we know what the behavior of the modes that get excited is, then we can use that to figure out where this transition is going to occur. And it turns out that you can actually analyze the problem to find these fixed points even when there are several excited modes quite efficiently. And that's something that Ben and Kiran looked at. So just to finish off, I just wanna show you some results for the critical coupling as a function of the number of modes, four cases where we excite the mode that's at twice the fundamental frequency and the mode that's three times the fundamental frequency. And the points here are plotted where there's a continuous transition and where we can figure this out using this stability analysis, the region here is marking where discontinuous transitions occur and we have to work hard to go looking for exactly where these happen. So for example here, with N equals eight here, eight modes where we're exciting the mode with frequency three times the fundamental, we have a transition here. But if we remove one of the modes, then we have instead a discontinuous transition and it doesn't occur until a stronger couple. So as we're adding modes, there are particular values that reduce the transition that facilitate it. And these are basically given by an integer times the mode number that we're driving minus one. And one way of thinking about what's happening, why these are special, is that as we add one of these modes here, we can then excite that and the fundamental using the drive, but also photons from the resonantly excited system. And so in terms of the unexcited modes, these as we get to one of these values, which is special, we can facilitate the transition because we turn on processes where we can draw energy from the resonantly excited mode and basically use that to facilitate the lower order process in the unexcited modes. Okay, so I think my time is running out. So I just want to leave you with my conclusions. I think these higher order processes are really exciting. They don't happen very often. And I think there's some evidence that they're really doing things that are interesting, something in between very low and very high photon numbers, where quantum effects are still important, which is unusual. And we're looking for an intuitive picture of that. We're also pursuing this multi-mode problem. And as you heard on Wednesday from Ben, one of the issues there is looking at basically what the quantum fluctuations are, what the entanglement is. In particular, Ben spoke about that below the threshold. But I'd like to stop now and thank you for listening. Thanks a lot, Andrew. So the floor is open for questions, Fabian. Hey, Fabian Hasler here. Thanks for the nice talk. I have actually a question to the first part and one to the second. So to the first part, it's like you introduced this delta as the coupling strength. Did you think about going to very large delta, at least in the theoretical analysis, because something special happens for delta equal 1? And the second question is like it seems in the second part everything seems to depend on this cut-off. And yeah, have you been thinking about making the cut-off a bit softer? OK, so the first question, I think yes. So the delta parameter is key, absolutely. And I think we've seen, of course, from Deni that you can have this parameter of order unity. And that really does suppress the phase space. And if you continue making it smaller, then yes, the phase space gets smaller. But these are processes that, because they're six-photon processes, kind of require a certain size to work properly. And so I think what I would argue is that if you make delta too large, then things actually become less interesting again in this case. So these oscillations, I think they're most sharply defined. And you can worry about what the spacing is and a value that's below 1. As you start to go higher, the first one is still there. The second one is barely there. And you wouldn't think that there was a sequence of them. And as you increase delta even further, you're really pushing this phase space down. And so these features, I think, are damaged, actually, in that limit. In terms of the multi-mode problem with the sharp cutoff, yes, I think absolutely. I think a smooth cutoff is certainly more realistic. But I think this is quite a nice place to start. And it's complicated enough. And I think it's a nice way to try and understand what these modes are actually doing. And then when you introduce a smooth cutoff, those features will get washed out. So for example, one thing we have done is increase the damping with mode number. That's one way to implement the cutoff. And what you find, of course, then is that basically as you keep adding modes, at some point it makes no difference, because those modes are highly damped. So yes, that's an important thing to do. But I think it's still quite fun to look at this in the slightly simpler case with the sharp cutoff and just see what's happening with these additional modes. OK, thank you. And I have a short question related to the second part, to the multi-mode. Do you think that you can understand all your results in terms of Flocke theory? I mean, all the multi-mode structure that emerge could be explained in terms of Flocke multi-mode resonances. You have a driven system, and your structures seems to me that it looks like the one that you can obtain from the Flocke structure. Yes, that's a very interesting point. Yes, I think that would be a good thing to do. I mean, at the moment, our main focus was really on trying to understand where these transitions occur. And so most of the modes aren't oscillating, and so that's not a problem. And then what you have to do is find what the state of these excited modes are. And we did find a way to do that. We haven't used the Flocke approach. If you, what we try and do is with the rotating wave approximation, just make essentially the simplest analysis where the time drops out for analyzing those six points. Thank you very much, Andy, for the nice presentation. I would very much like to discuss many of these issues with you. As you know, there is one, say, speculative question. Do we learn something when we consider the limit where the whole system tends towards a field theory? So considering the limit where you consider a field theory, do you think we learn something from this limit, which can, say, provide more input or other perspective than what you have shown? Yes, I think so. I mean, that isn't something I've tried to do, but I think the field theory limit would be interesting. And I think, I mean, more generally, I think these systems, they give rise to potentially very interesting nonlinear theory, field theories. So yes, I think that would be interesting. Sorry, maybe I can ask a quick question about these oscillations. Is there a correlation where I could see them in particular in a stronger way? Is it the spectrum or some other correlation in the system for the first part? Well, I mean, these oscillations are at present a very basic level in the most obvious observable, the average occupation number in the system. Or that striking quality. So you don't really need to go to something more complicated. The system is, classically, there's a bifurcation, but it doesn't tell you where the system is going to be. And what we see is that quantum mechanically, the system really takes a long time to decide where it wants to be. And essentially, it goes backwards and forwards. And that's so it's present really just at the occupation number level without going beyond that. I mean, of course, you can look at more complicated features as well. OK, thank you. Good. Are there more questions? I think not. And let's thank Andrew again.