 minute slots, so you should have like 15 minutes for the talk and I give you a signal some minutes before the end Can everybody hear me? Okay, perfect then first of all I would like to thank the organizers of this very nice conference for Giving me the opportunity to present our recent work, especially in front of a real-life audience and not a screen And I think this talk very nicely maybe continues, but we just heard so it's also about period tripling or higher-order period multiplication due to parametric down conversion in circuit QED and You will probably notice quite a few similarities also to the talk this morning by Fabian hustler Since this was kind of a follow-up project after that So because of that I'm also going to talk about parametrically driven systems and period doubling a little bit first And but I will start from a completely classical perspective so for period doubling we have a Demt harmonic oscillator that is driven parametrically proportional to the amplitude X of the system At twice the resonance frequency and then additionally There's also some small stabilizing potential that only becomes important if the amplitude of the oscillations becomes very large and as we heard this morning the system has a stable vacuum state below a certain amplitude in parametric drive where there are no classical oscillations whatsoever and then at a certain threshold the system has a pitch fault bifurcation where we get an instable vacuum state and To stable symmetry broken fixed points that correspond to an oscillation with half the driving frequency and And the exact shape and form of those points then depends on the details of the stabilizing potential What I want to talk about today is what happens if the system is driven at more than twice the resonance frequency so three times or higher and Mathematically the biggest difference is that instead of this linear parametric driving term in X now we have a nonlinear drive and this has quite a big impact on the qualitative behavior close to the vacuum state Because this means that the parametric drive is no longer able to turn the vacuum state in stable however, if the amplitude of the drive is large enough and at some point the Unstable fixed points that exist in the system will come closer and closer to the vacuum state and the value of stability around the vacuum state will become smaller and smaller and And today I want to talk about how quantum fluctuations can then induce a Transition to the symmetry broken fixed points We've already heard that For the period doubling case the quantum fluctuations kind of wash out the instability transition and we can already see photon emission below the threshold and Here in the case of period multiplication. They're actually necessary to even induce a symmetry breaking transition and Going along with the previous talks. I will show the same circuit again so one possible circuit that can Implement this in a microwave setup as a DC by the bias Josephson junction That's in series with the microwave resonator and then parametric resonance is achieved if a The voltage is biased in such a way That each tunnel in Cooper pair excites M photons in the resonator And you probably also seen this before this is the Hamiltonian of the circuit but in contrast to my predecessor, I will be talking about the quasi classical limit of small vacuum fluctuation strength which because it's a usually quite good to achieve in microwave resonators and be it's also a nice quasi classical physics But first I want to perform rotating wave approximation with respect to The driving frequency to EV over M. So depending on which resonance we are looking at and this then gives this Hamiltonian also with the Bessel function that we just talked about in the question session and In order to just study the transition from the vacuum state this whole Hamiltonian is actually not necessary for this We can just approximate the Bessel function to lowest order and just look at the dynamics very close to Zero amplitude So that Hamiltonian looks something like this, which I think is pretty much exactly what we just saw in the previous talk I just introduced this epsilon which is proportional to the Josephson energy and encodes the driving strength and This very generic Hamiltonian for the symmetry braking Has a face portrait purely classical that looks something like this So in the case of no detuning between the parametric drive and the resonance frequency The system has one single fixed point But this fixed point is somewhat special because it has a completely vanishing stability matrix This is the case of period tripling but basically looks the same for higher orders as well and This sounds very promising because this kind of means the vacuum state is effectively already pretty much unstable at least from a classical perspective However, as soon as there's even a small detuning between the Resonance frequency and the parametric drive this single fixed point splits up into M Unstable fixed points that form a valley of stability around the stable vacuum state and The distance of these fixed points depends on the detuning over the parametric drive and From this one could in a very semi classical picture Say that escape from the vacuum state due to quantum fluctuations is possible If the vacuum fluctuation strength is of the order or larger then this distance of the unstable fixed points from the vacuum state However, I haven't talked about what happens if we also include dissipation and Disappation is of course known for stabilizing the vacuum state So we definitely have to take that into consideration as well and Keeping with the quasi classical description of the system We can write down a lingerie equation that encodes the quantum fluctuations by this white noise Gaussian term here And for this system if we again look at the classical phase portray We see this is now at finite dissipation, but zero detuning so we can see the effect of the dissipation It still has M unstable fixed points that surround the stable fixed point at the center Now the distance of these fixed points will depend both on the detuning and the dissipation So indeed the dissipation does increase the value of stability around the vacuum state But from this one can still say that an escape is possible now We just have to take detuning and dissipation into account and go into the limit where they are both small when compared to the parametric driving strength and To make this a bit more quantitative We write down the Martin-Ciccio Rose action, which you might remember from Fabian's talk this morning Which encodes the exact same physics as the previous lingerie equation But we basically have done that instead of encoding the quantum noise by this stochastic variable xi Now the quantum noise is encoded by this auxiliary variable alpha q via this last term here and With this action we can look at the action that is accumulated when going from the stable Vacuum state at the center to one of the M Unstable fixed points and the action the minimum action that is accumulated then gives us the exponential escape rate By e to the i times that action and for the case of period tripling one can actually do that calculation And this is the exponential Rate that we obtain I think quite intuitively it depends on this distance that I talked about before so gamma squared plus delta squared Over epsilon squared is this distance of the unstable states from the vacuum state and then we have this Noise term here, which depends on the vacuum fluctuation strength kappa with this plus one Which is the quantumness of this calculation and the buzzer Einstein occupation and this additional factor two-thirds Is just what comes out from the Martin-Ciccio Rose calculation We have also done some numerical simulations of the longevity equation to compare it with this result So the first plot here shows the exponential dependence of the escape rate on the fluctuation strength and then the fitted Lines Then the slope is plotted here on the right and compare it with the solid line Which is the result here on the left and they seem to agree quite well After the system escapes from the state it will go to one of these Symmetry broken fixed points that I talked about in the very beginning that correspond to an oscillation of the parametric driving frequency over M and In order to look at those states a little bit more we expand the Hamiltonian from before with the Bessel function Up to next leading order To also take the stabilizing effects into account They are of course different stabilizing potentials and they can lead to very different multi-periodic states So often especially in the case of period tripling one looks at the Duffing oscillator as the stabilizing potential Which behaves quite a bit differently from this stabilizing potential here that arises due to the Josephson parametric down conversion So what a system could look like in one of these period tripled states in this case Is plotted here? So we have the current as a function over time and the black line corresponds to the original parametric drive and Then all these colored lines correspond to possible face locked solutions That are face locked to the parametric drive with different fixed phases and a certain amplitude and If we look at the again classical face portray now including this additional term Then you can see that this system in this case of Josephson parametric down conversion has six of These stable symmetry broken states Which is a bit different from the Duffing oscillator, which has three Which is the minimum that you can have because you have a three-fold symmetry in the case of period tripling and I plotted here the face diagram for delta equals zero which Which I will remain at for the last three minutes of this talk Because this is also the best limit in order to achieve this symmetry breaking in the first place And in this limit the system is equally likely to end up in any of these six states because each of the M or in this case three unstable fixed points via which this initial symmetry breaking is most likely to occur is Equally connected to always two of those symmetry broken fixed points So in the end, it's really completely equally likely to end up in any of these points and What's nice is that in the limit of vanishing dissipation or very small dissipation in relation to the parametric drive these six symmetry broken fixed points are actually completely equidistant around the vacuum state and They have an alternating frequency that is proportional to the parametric drive And then the last thing I want to talk about is the timescale of the defacing monster system is in one of these symmetry broken states Because one can really only talk about a symmetry breaking transition if that timescale is much lower than the timescale of the initial symmetry breaking and To do that we went back to the Martin-City Rose action and Again looked at the different tunneling rates from one of these stable symmetry broken points But in contrast to before where the vacuum state was equally connected to the three unstable fixed points and it didn't matter by out which one the symmetry breaking occurred because they were all Identical Here now the system has three different unstable fixed points and they all correspond to a slightly different case of defacing What we can see this is a numerical calculation of the different exponential escape rates via the three different unstable fixed points what we see is that at Slightly larger dissipation rate gamma over epsilon it is most likely that the system escapes via this green point and the green point actually always just connects these two symmetric Fixed points with each other. So this leads first to a defacing to a three-fold symmetric state before it defaces completely via one of the other unstable fixed points It's also quite nice. There's a very nice technique based on the Poincare cross-section method that allows one to calculate this particular limit here where gamma goes to zero and One can actually obtain an analytical result for the tunneling rate in this particular case It is also quite intuitive that in this particular limit the escape rate via these three unstable fixed points is Completely identical because in this limit the system performs many many oscillations around the fixed point before escaping So in that way these three points and the rotation trajectories that kind of connect these three points Make those points identical And yeah with this I would like to conclude with This short note that yes indeed the defacing time is much smaller than the time of the initial symmetry breaking if dissipation and detuning are small enough and That's what can actually talk about this period tripling as an actual symmetry breaking transition And yeah, but this I would like to thank you for your attention. Yeah. Thank you very much All right, there's already question in the back Can you please go back to the expression for the escape rate? For what for escape rate? I think it was It was No, no escape rate. Yes, and then you have our equals. Yeah, I wonder if it's valid at zero temperature Yes As long as the we're in the regime where the vacuum fluctuations are small then of course We just have this this quantum factor one there But then this is valid. Yes, because you have this like quantum fluctuations that cause the escape from the vacuum state We don't need thermal fluctuations for that Okay, thanks So it may be a follow-up question to that so if we were trying to make the Description more quantum and and we were for example chirping the frequency to go through the delta equals zero at a particular M Would it be possible to make? some sort of cat state or voodoo state by splitting the The trap state into all these different so so basically like it's like a yeah superposition You're competing with the phasing and the question is is it feasible? I mean, of course, I've only looked at this semi classical regime. So there are no quantum superposition is really possible It's more of a Statistical mixed state then that the defacing causes So I think in order to really look at that one has to take more quantum effects into account I think it could be possible But I haven't looked at that in more detail Thank you Maybe just a very quick follow-up on that question isn't isn't this what has been done in the group of Michelle devoree where they looked at the to photon pumping where where you split these vacuum fluctuations in into two and And then if you measure fast enough, you see that there's coherence actually Yeah, I think for for two it has been done I'm not 100% sure if it also then works for three or four Because there is quite a bit of qualitative difference between the different cases, but yes, of course in this way It sounds possible. Yes Thanks. So another question Yes, sorry if it's a silly question if I were to measure this stability diagram I just need to to sample the output field many many times construct some histograms and I should see three blobs Okay, yeah, or six or whatever. Yeah. Yes. Thank you All right, so last question and maybe the next speaker already Start to connect My question is about me to calculate this are you use a settle point approximation, I guess So, yeah, it's really just exactly my questions. What are the corrections to the subtle point approximations? to this, I mean you're saying this are is Is is valid up to temperature zero and then my and it's of course only the exponential rate So there are three factors that I can't calculate That I haven't looked at. Okay Okay, then thank you again for the talk