 Yeah, it's just that it's okay. So let's start I Want to thank the organizers for inviting me here. It's a beautiful place and It's a great conference after the pandemic to meet everybody all of you again and so What I want to tell you is actually about microwave photons, but it's it's even a bit more general So we gave it the very broad title the universe reality of photon content below a local bifurcation threshold but I will explain it what it means in terms of microwave and Most of the work has been done by Lisa aren't was also in the audience. So if you have any questions, I most likely will refer to her Okay, so this is very old physics what I'm going to talk about and it's all that there's actually commercial You find on Wikipedia a new amplifier a battle snows. It's not so new anymore. It's It's like I don't know 60 70 years old But yeah, so Bell labs figured out that there's certain kind of amplifiers which have much better with the noise And some of the reasons is that it's actually energy conserving So you can actually implement it also as a quantum process But what it does it changes the frequency of the radiation? So in our you know our view of actually looking everything in terms of photons But this device does it takes one photon and produces two photons out of it And so there are different cases how you can do this and in one case Which is called degenerate is you take a photon and you split it up into two photons with the same frequency That's called degenerate And then there's non-degenerate cases and the non-degenerate cases is the same process But you split it up into two photons with different frequency Now you might wonder what's so special about to you could also do three or four something and there will be talks today about this Other cases for my talk that we be actually restrict to the two photon case And so the question is what happens if you ever have a process like that And that's actually I mean on the classical level is something which is called Spontaneous oscillation, so I'm just looking at the degenerate case Yeah, so I take one photon and produce two So the classical equation of motion which does this is a harmonic oscillator and then it's parametrically driven And so you change the resonance frequency in a periodical fashion and the simplest case We're actually want to produce out of two omega single omega you actually drive it with two omega So that's the right frequency and if you solve this equation and you look at the stationary state What you actually see it plot your driving strength and here the amplitude that you know The system has a solution with no amplitude and that's a very natural stationary solution because they're stamping on top And then you start driving stronger stronger stronger stronger and at some point it spontaneously Oscillates and they're actually two solutions one with plus and one with minus and there's a certain phase locking and this part actually is classical physics, so we're not interested in that and So what we interested is actually what happens down here, so this is the sub threshold And there's stuff also stuff which you heard before in the top before there's a quantum emission of photons squeezing Entanglement all that stuff happens down here So for my talk we actually concentrate on this region below and we actually try to stay close to the threshold But there's a little window Where we're not allowed to work in and this little window I think it's very interesting, but for this talk will actually neglect that so that's where nonlinearities are important and The question for this talk is what is actually the photon counting down here You have a system like this and you try to look at the number of photons that comes out of the device And cheaper and is actually looking at me because he knows that this has been solved a long time ago It's actually his work and which I now present So the whole thing has actually started with an experiment where I think some people in this room are actually part of So this is an experiment in Sackler where actually here's the voltage and the voltage actually acts in this chosen devices So the device is such you apply a DC voltage here for chosen junction You have a harmonic oscillator and this chosen junction is actually the parametric drive on this harmonic oscillator And in chosen junction in such a way that actually the driving frequency is set by the DC voltage that the AC chooses an effect And so this is very easy because you can actually to easily tune the driving frequency And so the frequency of the oscillator is fixed and then at some point You know you get kind of a mission of light and then if you double the voltage you get also a mission of life And that's this spontaneous conversion So once we saw this we actually tried to do some theory. So it's with you Lena sorrow from chiprion And we actually looked at the counting of the photons And now you know given the last talk we have to be very careful what we mean by counting So so the idea what we have is actually that we Take a time window which is very very long and we accumulate all photons which actually arrive in this time window And that's our observable So it's not time resolved in that way and and how you count the number of photons I don't care you can count the energy that has arrived over the full time or Whatever there are different ways of actually doing it but you take a very long time window and The reason why you do that is because since stuff becomes universal And what we actually figured out is that there's kind of something which is called the counting statistics Whose derivative gives you the cumulant? The cumulants for you is just the first cumulant is just the average number of photons that arrive The second cumulant is the variance and then it's kind of generalizations of this idea So the counting function actually tells you about the cumulants by just taking derivatives And what we figured out is the counting function has this very simple and neat form So this is kind of the damping. This is the measurement time So this is just a number of photons and all cumulants are proportional to how long you measure that makes sense And here the delta actually tells you how far away from this onset of spontaneous oscillation You are so this how close we are to a threshold with your drive. Usually you'll be low and Kai's the counting function Okay, so this was a calculation back in the days. It's already ten years ago And so what's the story the news story now? Yes, so we looked at this case So that was the case where we drove with twice the frequency and now comes two photons with half the frequency and Then some years later. We looked at this other case and on the general case So that is you drive and then you have two oscillators with two different frequencies which add up to the full frequency And and you know historically, this is wrong This is a paper that came out this year, but yeah People also look at micromasers where you drive a current through that and then their photons emission this transmission lines You kind of tunnel coupling your double dot system So there are different systems you can do the calculation and all the system have in common That you start driving and you drive stronger and stronger stronger and at some point you get this classical bifurcation and the question is about the phone statistics below that classical bifurcation and What turns out is the answer is always the same It's always the same answer So now this might look surprising to some of you or it might look not surprising But that's just a fact. Yeah, so the parameters. This is a microscopic parameter Which just tell you you know, what's the probability? What's the rate of photons in this line the deltas how far they tuned you are and the guys just It's always the same function and You know the systems are sufficiently different that you might wonder what's the reason why this is the case and what's the theory behind that And that's what I'm going to tell you So we try to extract universal physics from this and I also wanted to show some experimental results So this is something which I got this is from optics Yeah, so did they actually look that at a spontaneous down conversion Anticagulated something which is G2 which is a bit more information than what you shown before G2 It's just a probability you find a photon after you actually detected the photon before so it's a correlation of photons And the G2 and all the systems look like this Has a peak and it has those kind of two parameters one is the width of it and the others the height and so What you realize is that there's one parameter which is the correlation time Which is the width of this peak and the other is kind of the number of photons in the cavity That's kind of proportional to the height It's not really the height, but it's proportional to the height and they are dependent on a single control parameter Delta You don't have more control parameter if you're close to the threshold And then you realize that actually both of them diverge close to the threshold So the the correlation time diverge that's just a hallmark of a non equilibrium phase transition Because above the phase transition when you just don't relax to a single state anymore That's why you suddenly produce two states two equilibrium states were before you had like one station And the second is the number of photons you kind of get more and more photons in the cavity until you produce these Coherent states which corresponds to the spontaneous oscillation about the transition now if you have Both of them diverging that's really the perfect place to do a classic classical description of the problem You have many many photons and the dynamics is very very slow. So you can do classic classics So that makes it already much much much more simple no because I mean for me quantum mechanics is always this complicated theory Something which is quasi forget classical. That's that's kind of simple And then we can connect it to something which has been studied very very often, which is the dynamics of of of So so what you know one is the equilibrium phase transitions And then you look at the dynamics of all the parameters closer equilibrium phase transitions and which we what we do in this talk is actually connect This non equilibrium system to the theory of dynamics in equilibrium phase transitions Good, so I mean I just give you an example So what I claim is that you know the physics here close to the phase transition But below is actually described by something which is called the Glauber model Which is in Hohenberg helping classification is called model. Hey, okay? I don't know who came up with these names, but that's how it is model H That means also the most basic model good It's good So it's purely dissipative dynamics in the rotating frame So I write down the equation of motion for the amplitude of the oscillation So I mean it's kind of damped down and there's some force applied due to the fact that I'm kind of driving And then there's noise And the noise is just there because of fluctuation dissipation theorem and in the rotating frame It has this famous factor 2n plus 1 and I would claim this plus 1 is the only place where quantum mechanics appears in all my Okay, and and you know this this is a stochastic Classical differential equation. There's an equivalent description in terms of actions Which is called Martin's teacher Rose action. This is made for the experts So so there you replace the noise with something which is a till day a second field And that's actually conjugate so kind of get quantum mechanics in this classical stochastic systems But that's not so important for my tongue And then the question is how you count yeah and the counting is actually quite simple simple first The ordering of the operators doesn't really matter. I mean it took us some time to realize that but Ordering doesn't really matter. You have a large number of photons So normal ordering is completely irrelevant and then the amplitude squared just gives you the number of photons There's some kind of conversion factor And and the alpha which appears here is something like the fine structure content I failed to point out that's kind of the characteristics impedance which you have in your circle Which compared to quantum impedance which tells you how to convert So you have this action here, and then you add just another action which which accounts for the counting and that's your complete action And you know, maybe you're saying yeah, okay This is magic. Yeah, and often physics is especially theoretical physics is magic But you want to connect it to kind of basic fundamental Physics so you have a system in mind and this you can do and this will be the most technical slide of my talk So if you really really want to describe this system Yeah, you will start with some quantum mechanical evolution of an you know of an open system Which is density matrix dynamics? You will have forward and backward propagation. That's what density matrices do Then for example, I mean you can do different things you can do lymph blood and stuff But we we tried to do all this kelvish because it's easiest to do the semi classical or quasi classical limit You write it down in terms of a path integral where once describes this forward propagation the other side described this backward propagation So just give it a plus and a minus and then these different components have different terms So for example chosen junction is something which one calls Hamiltonian dynamics And it's just on the forward contour and the backward contour. It's just minus of each other It's just what what these Hamiltonians would do here and Then there's some other terms which describe this Damped harmonic oscillator, which is essentially Caldera like it Okay, and Then we have to do rotating wave approximation. This is clouber dynamics was in the rotating frame Yeah, you have to do rotating wave approximation. So we describe a complex amplitude And then we do a semi classical approximation And that's the fact that there's low dynamics and large number of photons, which means the plus equals approximately minus And and then everybody who has worked a little bit on this already realizes how the market teacher rose action comes out You expand in the difference between the two And then the sum of them is what's called classical That's just the amplitude and the difference is this till the field Which I introduced before so that's that's the simple way how you would derive this which I first just presented Result and You can do this for each system again And what I'm telling you the action that comes out in this limit will always be the same because it's essentially constrained by symmetry There's no other terms which you can write down in terms of symmetry Okay, so now we settled on the on the model the question is kind of what kind of force we put in there and The forces you can put in there is also pretty Classified and universal yeah, so we had this damping. We had this fluctuation dissipation noise Which was just the fact that we radiate out of the system The question is what kind of force acts on our system and the forces are also kind of classified Yeah You can kind of you know at this onset of the instability threshold The amplitude is small you kind of do a Taylor expansion and then at some point you stop No, and you should stop better at something which is nonlinear and then the question is which terms you take so the x squared You can actually remove by some proper Shift of variables and then you have this model in general and that gives you just this cast Catastrophe and you can read up in this this book And so what we were looking so far is just when we when we have an additional symmetry Between the plus and the minus amplitude and that actually constraints our be to be zero that would just be we move here Really on the on the zero line through the cast that what what we were looking and then you get this pitch board by vocation And we were looking at this region down here In general and I will explain you later We could apply additional a coherent drive Just a coherent current on top and then we would go here would add this term be on top And then it looks different. So here is like one stable state Becomes into one unstable and two stable states. Here's one stable state continues And then there's an unsteady stable separating from something else So there's a high amplitude oscillation a low amplitude oscillation which appears simultaneously And this is called fold because it's like folding a paper and this called pitch for because it looks like a pitch fork Okay, so now we have our model together. I mean there's just one step which I would like to do So instead of a tilde I multiplied with alpha. Yeah, such as accommodation relation is not one but alpha and And you know, then the action has this one over alpha in each of the terms That's nice. So every theoretician sees an action with one over alpha in each of the terms No, it tries to do small alpha expansion and then you just limited by the saddle point so we can actually, you know, look at the different cases and kind of This is counting this is the dynamics and we can do the the saddle point dynamics And what we actually see is that it's essentially classical dynamics with a kind of a strange Hamiltonian and then classical dynamics We all know we had our classical mechanics lecture So it's kind of a funny Hamiltonian and then you can look at it at settle points and what you actually see at the Pitchburg everything is dominated at small amplitudes Which is also obvious and then and then the whole picture looks a bit more complicated for the fold But I mean this integral you can do it's just technical and you do it in the saddle point approximation You do the integral and out comes the cumulon generating function So what I claim this is the Universal cumulon generating function for the statistics close to one of these casts There's the first term which we've seen before which was essentially also known in the literature and then there's a second term Which comes if you kind of you know break the symmetry of the cusp That goes with B, which is kind of the deviation here from from this cusp This squared and here it there's a fine structure constant inside Otherwise, there's only Delta. Yeah, and of course the detuning And a so the a detuning it's just included in the Delta Okay, and there's a little little little Rishim which you are not allowed to use our theory because the fluctuation becomes so large that we actually Are not allowed to do this in a classical approximation And then you can actually calculate the cumulon and how you do that you can ask Lisa I don't know I mean there's kind of double factorial and stuff like that appearing It's complicated, but yeah, so it also has two terms And the first term corresponds to the pitch fork and the second term corresponds to the fault and they just add up in the cumulon And and what you actually have if you're in a generic place here, you actually have a crossover here So for largely tunings The pitchfork Dominates if you're somewhere here You can feel mostly the pitchfork and then for smaller detuning then you start feeling the fault And and then that's very very small detuning. You're not allowed to use our theory anymore So that's the way it goes and this is an expression and you see that actually a small fine structure constant You can really separate this regime And the theory is anybody only valid a small fine structure constant Good Now What you can actually see here is already the fact that there's some kind of divergence of all the cumulons When you get close to the threshold that's also what what is expected here at some point It will not really diverge nothing diverge But that's this little shaded region which are not allowed to use the results But otherwise it's actually divergent and and the way it diverges is very much similar to critical exponents in equilibrium phase transitions Yeah, this this this exponents are universal for each of the cumulants now. We have more than just second cumulants Yeah, equilibrium phase transition. They're always Gaussian here The stuff is highly non-gaussian and Okay, I have to click through and you see that actually the divergence of each of the cumulants is actually given With a certain critical exponent and and they are different for the pitch work bifurcation and for the fault bifurcation So whether you have additional the symmetry between plus and minus amplitude or not Actually changes the critical exponents, but it's not depending on the system, which is on the basic of it No, I mean I've shown you three systems, which all show the same critical exponents and Then for example, you can look at the funnel factor and this is quite interesting because the funnel factor tells you how many photons are correlated with each other and Both of them the funnel factors is also diverging and what it actually tells you if you look at it in details Is that they're more photons correlated that they're in the resonator at a single instant of time Which is maybe counter-intuitive, but the whole system tries to get to get into this coherent state So it loses a few photons and new come in and they kind of fill this coherence, which is already built up So you have more photons correlated than they're in the in the resonator at the single instance of time And the other thing is this is actually, you know, this would be the dynamical critical exponents Like how does the time scale diverge with your parameter in the system? And that's actually one in both cases. Yeah, and we believe it's kind of quite universally one This is just I mean you have the photon current and look How long you need to actually produce this number of correlated photons Then there's other stuff which you can look which is universal one of it is rare event statistics I told you it's completely non-gaussian the statistics. Yeah, so the first two cumulants don't tell you Everything and we all know that measuring higher cumulants is kind of complicated What often is much more simply is actually looking at extreme events Yeah, you kind of do a threshold detection and see when when when I don't know Your signal actually goes beyond a certain threshold and then this what you see is actually probability to see certain extreme rare events And it's actually logarithmic. Yeah, so it's the gaussian is just this parabola The dotted behind and the fold bifurcation is twisted towards the higher And the pitch work even more. Yeah, so they're more and more twisted. You can also see it at cases on in the pitch for a case You actually essentially don't get any probability for zero signal Yeah, whereas in in the fold case you get the constant probability for zero signal So the kind of more twisted towards the side and the same thing on the right side on the right side This effect of four in the logarithm. Yeah, so it's Okay, so about potential implementation. Maybe that's of interest to some of you So in this community again metaphysics is very generic, but in this community This is the best setup which we could have come up. Yeah again DC biased Josephson junction a resonator and some detector somewhere the alpha which we had is just The fine structure constant is just the characteristic impedance should be small and Kind of we want a high quality resonator Then the parametric resonance is set by the voltage so the voltage sets us the frequency here And we have to set the voltage equal to omega such that one Cooper pair produces two excitations in the photons and and then the driving strength is is just this chosen energy which you can tune by split choices and junctions and Now the other axis is actually an additional coherent drive And you have to apply it properly that it's actually in phase With the oscillation that wants to be built up. So you have both axis under control and then you can measure and you I'm actually already finished even though the The chairman says I'm five minutes left, but that's good So what I actually showed you is that there's a generic characteristic function of the photon counting and it's based on the universality of You know Hohenberg help ring classification and we believe it's actually can be generalized in all kinds of directions I mean, you know these things can be done in one day, which would be chains of Of these kind of devices The fault in the pitch verification lead to different critical exponents and different fan of actors But can be tested in a single device So this whole thing can be done in this device and so we propose a concrete microwave setup for more information You have to read this paper or talk to Lisa or me and I just want to announce something this Blue region, which is very very narrow is highly interesting and we believe it's also universal even though it's actually less I mean here this thing is actually extremely universal So the many when nobody would see a degenerate and a non degenerate amplifier will produce the same statistics So we believe that there's also some universality in this regime and we're working on that But we don't have results yet We have good indications that will work out and there will be a paper. Hopefully this year coming out So if you're interested that you have to wait for that Okay, I want to thank for your attention So my question is about rare events So I wonder to what extent the statistics of rare events is universal because your Universality is based on an expansion right and if you have a rare strong event Then you are beyond the the expansion you're absolutely right and that's maybe even something which we would have to refer No, I think you're absolutely right if you kind of go far away at some point you expected other physics set in yeah But what I'm trying to claim is that as long as the fine structure constant is small That this it's kind of a crossover Thing so as long as the fine structure constant is small then these non linearities or something else Set in a very very far point and if you kind of at some point you see a deviation And if it's too early you have to kind of lower your characteristics impedance and we can figure out at which point it sets in But yeah, we haven't done so so two questions one is about Lazing and synchronization there there. I guess they have they start out with a slightly different system in the sense that they're driving maybe not always symmetrically and there they talk about Looking for complex eigenvalues and maximizing well minimize Minimizing the loss and an imaginary time sort of perspective and leading to synchronization So it seems complimentary to to this and I was wondering if there's Maybe a deeper insight and the second question was about the rare events and what would be the Is there something non-Galcian and in the quantum Squeezing that we can measure what what would be the signature exactly there? So one thing is yes in optics. They use mostly link-laden formulation and there's a mapping and we can figure it out and And we believe the lasing transition is is actually also in this universality And there is for example here this curtain is and then then keeling they looked at the dickie transition And it's the same Okay, so I think the statistics a wave away from the blue region will depend much more on the detail It's still universal but less universal, but this kind of Further away is kind of extremely universal. That's my feeling. It's kind of we were sad that it's so universal But yeah, and the second question was about the non-Galcianity I mean what you have to really detect and you know, it depends on your setup or you have to detect it that it's not I'm not sure in which direction I'm going You have to detect you see it's a logarithm of the probabilities You have to really detect rare events and you have to figure out whether it's a parabola that's Gaussian or it's not a parabola and So best is kind of detect a few events up here to get the curvature I mean the coverage you can also get via noise You do a noise measurement an average measurement Then you know the parabola which should be and then you try to detect some points out here and see that It's not on the problem by that you kind of show non-Galcianity of the thing thinking in the analogous with the statistical physics when you are talking about universality you usually Get universality from the symmetry of the model that is broken at the transition Can I think in this way and if this is the case can I think that you espon out the mean field exponents? So first mean field exponents in some sense These are the mean field exponents of the non-nuclear equilibrium phase transition. That's why it's maybe so universal What we are doing is essentially the mean field theory because it breaks down close to the To the threshold where fluctuations become important. Yeah, so that's why we're working on the other part the other thing is that it's it's kind of universal and This model a is extremely universal I mean so if you go through the fault then that's that's without symmetry essentially Yeah, that's essentially without symmetry and if you go through here you have this additional symmetry that You know plus amplitude the minus amplitude are the same. That's forces you to go along that line So there's kind of one symmetry But you're right. We're doing just mean field, but it's non-equilibrium. So I claim it's a bit more complicated Any other question so in the experiment Very you saying that they would need to be able to count the number of photons in there in the last experiment You said like in in some time interval or is it enough just to you can't see a pee pee of I was that just was I Intensity I I said photon currents the number of photons per time which you can count by photons or by energy or by You know, whatever you can also measure the quadratures if you want I mean, you see the fun thing is if you count for a very long time it essentially doesn't matter How you count it? Yeah, you have to accumulate all the signal over the long time Nothing specific here. Yeah, the eye is just a photon current because the number of photons will be proportional to time You divide them by each other you get the eye. I call it photon current Yeah, so basically it's just the power It can be power as function of time power. Yes, and the power of course has some kind of normalization Which is the energy of the mode Okay, so you don't necessarily need a volometer, but you could measure the volometer But you would also measure it from the quadratures. You can measure it by the quadratures No, at some point we entered the stuff which I have been discussing with many people like Do we need a measurement theory for all this? I Don't know. I haven't seen anything that we need more than this kind of Keldish something but if you measure something which doesn't fit with the theory then we have to look more carefully Right. I guess you mentioned this already, but I'm just thinking that for example our volometer has quite a lot of noise I mean the naturals and it's less than you know Usual microwave photon that we are we are looking at but on the other hand I guess you said that if you have noise on top you could kind of try to just measure the noise and you know There's so many photons coming out the reason why I do quasi classics is because it's essentially tons of photon coming out So we really you go, but I mean you have to design it in such a way You're not going to detect single photons necessarily, right? But you have to accumulate the number of photons over time So so in that sense it would be good for a volometer not to lose the energy too fast But you kind of accumulated or something. Yeah, I'm just thinking about the width of that distribution if I have noise Yeah, if I have noise on top and if they have the noise is broadening that a lot Yes, but you see the width is given by the average signal Okay, so make the average signal larger Could you just explain you said on the other side of the bifurcation? Classical it's not interesting Yeah, the threshold also has I mean there's an interesting physics on both sides of the threshold Yeah, so this this blue dashed line comes below and above but then at some point it becomes Poissonian or whatever. Okay, so I think we can thank Fabian again