 online, you have as much time as you want because you won't see me waving at you. No, I'm kidding, of course, people are angry, I'm kidding. So please, Thomas, the floor is yours. You have 15 minutes presentation and maybe I will try and wave at one point. Thanks. I see. Thank you, Niko, for the opportunity to participate in online this conference and also to the other organizers. My talk today is about a proposal for an implementation of a political just an ongoing way parametric amplifier. And this is an in collaboration with Alvaro, with Juanjo, Alejandro, and Diego, all at Madrid, my same institution. Well, I start by recalling that microwave signals coming from super conducting quantum secrets are typically very weak. And therefore, one requires an amplifier to detect them. And in this quantum amplifier process, there is always a gain factor G and some other noise end by the amplifier. And therefore, we also require isolator elements in order to protect the quantum systems from the noise coming from the amplifier. OK, one of the most advanced amplifiers so far in the market are just an entire way parametric amplifiers, which are built of arrays of just some intensions as shown here. Using this special engineering, this device has shown a very nice performance like with gains above 20 dB, with a noise that is near quantum limited and a very large bandwidth on the other gear heads. And of course, this device works very nicely, but will still even better to make an upgrade to them and make them to amplify non-reciprocally. This means that if you send a signal in the input, the signal is intellectually amplified from left to right without any reflection and also without a backward noise. And therefore, we can avoid having these isolator elements in the circuits and we can integrate the quantum system together with the preamplifiers quantum amplifiers in the same chip. The problem is that this is not so easy to do. And so far, this has been realized experimentally only in narrow bind devices, like as shown in these references. But in this talk, I want to show that it is indeed possible to do a broadband and directional amplification using the concept of topological amplification introduced in these seminal papers. Here, the challenge is to engineer artificial gauge fields and non-local pumping terms in these kind of devices. But I want to show that this is possible with the current superconducting circuit technology. OK, so the implementation, the proposal looks like this. On the left, I show the superconducting circuit scheme to realize this amplifier. And on the right, the quantum optics model that is realized by this circuit in an abstract form, let's say. So it has three main ingredients. Our proposal, the first is in blue. This is a just an induction array, where just an induction that are depicted here with crosses. They induce cross-care couplings between sites of this form and also self-care couplings on each on every side. And capacitors induce linear couplings between sites J8. And the second important ingredient is to couple each side of the just an induction array to input output ports. Not just the first one to send the signal and retrieve the output signal at the final site, but actually in all of them. And this is because we need to have a constant or a homogeneous decay on all sides to stabilize the logical phase that I showed late. And the third and last ingredient is to couple this just an induction array to a linear array of superconducting linear resonators as I show here, which have just standard or linear capacity hopping between sites and also it couples capacitively to the JJ array. Okay, and the trick here to make the system topological is to do the four-way mixing via the linear array as I show in more detail in the following. So the key here is to break the time reversal symmetry in the JJ array by driving on the auxiliary array and engineering the boundaries of this array. So we have input ports in the auxiliary array only in the first side and in the last side. And we have three conditions to break the time reversal symmetry. And the first one is to drive strongly in the first side here, V1 on resonance. And with this, we will put a lot of photons in the auxiliary array. Okay, the second condition is to have a very off resonant coupling to the other array. And because of this, all the photons that enter in the first site, they will travel to the right on the auxiliary array without jumping to the other just on the red part. And when they are approaching to the other boundary, we set the decay of this antenna here or this transmission line gamma to be exactly equal to the hopping JB. And if we do that, there will be a perfect impedance matching on the boundaries and all the photons going to the right here will leak out without any reflection in this auxiliary array. And because of that, they will be in steady state, they will be a classical running way built in this auxiliary array moving from left to right and you will have a well-defined phase five equal to pi over two in this case. Okay, and this is very important because this will induce an effective drive so that we can eliminate these auxiliary cavities and the effective action of this is to induce a gradient drive on all sides of the JJ array that has constant and homogeneous amplitude and still has this phase five pi over two coming from the auxiliary array. Okay, and the trick is to use this effective drive to do forward mixing on the JJ array. So we have a strong coupling and we can displace in this forward mixing, we can displace locally all modes of the JJ array here in this form. We have some mean displacement alpha and some small quantum fluctuations around this mean field. And what is very nice is that the same phase five coming from the pump actually provides perfect phase matching for the JJ array even without doing dispersion engineering. So actually here the system is completely homogeneous and assumed to be homogeneous. And as a result of this, the mean field displacement also will be homogeneous and given by the still order equation in steady state that can be solved numerically in a very simple way. Okay, so now regarding the different patients when we have a strong pump, we can, the dynamics becomes mainly linear and it's given by this one to land demand equation where it has a local decay kappa on all sides here given by this term and in the gradient part we have a parametric amplifier Hamiltonian which has two types of terms. Actually the first one is proportional to this kappa are parametric pumping terms like two photon driving terms in the system. And they will have two types of this term. One will be local parametric pumping terms proportional to JS on each side, which will be coming from linearizing the self and the cross-carnal linearities and alpha square is the number of photons in the pump. And also will be a non-local pump JC here coming from the, for linearizing the cross-carnal linear. And in the, there will be also photon conserving terms represent to M, which will be just standard and onsite the tuning and hopping with the sides and very importantly this hopping is complex. And the same phase coming from the pump that I showed you before enters here as a artificial gauge field that breaks the time reversal symmetry, the dynamics of these locations. So this gauge field in combination with the constant decay kappa and with the non-local pumping term JC allows the system to go to a topological steady state where there is directional amplification. This means that photon centering here will actually get amplified from left to right and will be accumulated on the left side. This I will show in the following. And to describe this in more details it's very convenient to introduce this non-emission dynamical matrix which come from writing the same equation in a matrix form. Okay, so this non-emission matrix contains all the co-edients and the dissipative couplings and allows to describe in a unified way the spectral and the logical properties of the system. That's why it's very useful. Okay, so the Green's function matrix which is basically the inverse of this non-emission matrix describes all the amplifying properties of the system. So for instance, the gain if you send a signal in the first site and you retrieve it in the site J it's given by the modulus square of this function and the reverse gain. So for a signal propagating in the opposite direction this has almost the same form but it depends on the Green's function with the reverse order ij, you see? So from here already we can see that if we want to get non-reciprocal amplification the Green's function must be asymmetric basically, okay? But for what kind of parameters in our system we can get this asymmetric Green function. And for this we use a connection of the open quantum system that is our amplifier to a topological band theory, okay? This was developed in these three papers in our group. Basically what we do here is to construct an extended Hamiltonian, this admission from the non-admission Hamiltonian in this form, okay? So this other h is by definition admission. So since it's an admission we can apply all the, we can do it as a, let's say it can be in some cases a topological insulator system and we can apply topological band theory to it and telephone way in a standard form. So we can diagonalize it and when this extended system is in an topological non-trivial phase for instance for these parameters of our system then the system will present zero energy modes and a topological inside this topological band gap, okay? And associated to every, these non-zero modes there will be states actually that are exponentially localized on the boundaries of this kind of artificial system. But this implies that we show in this paper that in the physical system in the real amplifier this means that when this happens this means that the green function has this exponential spatial dependence respect to the first side. So they depend on this where this theta is the correlation length of the edge states. And this implies that if we, that the gain and the reverse gain will be exponentially enhanced when we have this topological phase in the system and the reverse gain will eventually attenuate. So this is what we call topological amplification which is non-reciprocal but also exponentially enhanced, okay? This is the essence of topological amplification and now I want to show how this, well how this performs in a real experiment with welding circuits and kind of realistic parameters, okay? So here I showed the gain and the reverse gain for amplifying a signal frequency close to the pump, okay, a little bit away from the pump. And here we can see that the gain actually increases nearly exponentially from left to right as a function of the number of sites in the array. We can see already for both sites we can reach almost 40 dB of amplification. And for attenuation it's also exponentially suppressed this is in dB up to less than minus 40 dB. And for more or less realistic parameters the problem is that I haven't showed this yet but the gain increases exponentially but also the noise. So actually we need to stop at four sites only because then to avoid the saturation of this device. So for these parameters that we choose like say 60 minus 61 dBm of pumping we get 280 photons, okay, in steady state in the array and this should be much larger than the fluctuations. So we need to stop at eight but still we can get 25 dB of amplification and minus 35 dB of the say isolation in the other direction. And keeping these eight sites we can now study the bandwidth response of this amplifier and also the added noise as a function of frequency which is basically the total noise added by the amplifier as a function of frequency divided by the gain at that frequency, okay. So the gain as a function of frequency is in blue and the other noise is in red and in blue we can see that we can reach this 25 dB amplification close to the band frequency but still we can get a bandwidth on the order of 2.2 times J which in this system which is J is 0.8 GHz for the parameters we chose we can get almost 2 GHz of bandwidth for more than 20 dB of amplification and regarding the noise for the same frequencies the limit here in our convention is one for the other noise so we can get 1.3 so almost near 1.2 limited amplification here and the central frequency is tunable because if we put tune for example by external flux the frequency of the auxiliary arrays we can tune this and the volume is 5.5 GHz. Okay and the last I want to mention is that since this topological amplification direction amplification has a topological origin it is proposed against a disorder in the system we assume it is homogeneous but can have some disorder so and this protection comes from the final symmetry of this extended Hamiltonian that is always present by construction so it does not depend on the specific parameters of the system and therefore the system is actually robust to all parameters as we show here so here we calculate the other noise and the gain at the last side as a function of the noise in all parameters in the phase, in the tuning, in kappa, hopping and the pumping terms and we can see that almost at least for 10% disorder this almost doesn't change from minus 1.5 to 1.4 basically and even in some parameters can be even 30% disorder and the gain is similar and even for certain noise the gain can increase and always the gain can reduce so only slightly even for 10% and this kind of phase has been seen in other topological systems okay that was all I want to tell you and so in summary with this proposal we think that implementing a topological doesn't have any way parametric amplifier is within experimental reach with this kind of setup the key to make this topological is to couple the array to an auxiliary array and implement a gauge field and a phase matching via this I've shown that for more or less realistic parameters we can get a good gain and good reverse gain near quantum limited noise and bandwidth of two gigahertz for only eight sites actually can be very efficient with you don't need a very big system actually and can be homogeneous without so much non-local engineering and it's supposed to be solved and this will be very soon available in the archive and for the outlook I want to comment that if it's possible to build this kind of device one could get rid of this blue isolator also because the system is small let's say on the other 10 sites one may be study the saturation effect from first principles like incorporating the non-linearities in the system completely and also that the system can also be interesting from a many-body perspective and one can study imagine state-to-body effects in the system for instance with different winding numbers and so on and as I showed here as an example of another prepping that we will put soon in the archive and sign all these features so thank you very much to you for your attention and to my collaborators in Madrid thank you I see many hands I think Denis was the first so hi, thank you for the talk hi in this type of two paths with two lines you have generally one classical line the one in red in your last slide and on this one you pump and on the other one you put your signal and you of course because it's four-way mixing you have idlers and the signal and the pump make your first idler then the pump and the first idler make a second idler the second idler and the pump make a third idler and you have a series of idlers like this that can get the power from the pump how many idlers have you considered in your calculation? how many orders of idlers you say? yeah yeah I think we do the calculation in real space to say so in principle I think all the orders are taken into account so at least when Julina writes the system okay so the gain that you quote is with all idlers taken into consideration yes exactly, so all idlers they enter in the noise because we are operating the system as a phase in sensitive amplifier and the idler at the end enters in the noise part the idler at noise thank you welcome next one is Carlos hi thank you for your talk if I understood when the impedance matching of the red line provides both the k-reality to the system but also the phase matching condition for the traveling wave so how sensitive you are to this phase matching condition before you are losing the phase matching on the blue line and then the amplifier stops working I see so in principle how sensitive you say so we assume that what is very important is that the the gamma here the decay in the last side has to be equal to the to the hopping in this, the linear hopping in the array but this actually we can show that when this happens we can get an exact solution of this running wave that appears and if you are away from this let's say you have a small disorder a small perturbation then you will get a small change, a small difference in the file in the phase so you will get pi over 2 and some small disorder let's say but then later at the end we phenomenologically change this disorder by actually more than 10% and we show that the topological phase is robust because from the topological protection so you would say that as long as gamma and JB they are always in a 10% window similar? yeah more or less maybe even a bit more but 10% window I think is yeah for sure it works yeah okay I think you have another question as well a very quick question I didn't really understand if I have to pump independently all of these eight nodes uh no no just you pump the the first side and the B and the the the same chain here will will be behaved as a way guide that this will distribute the pump to the A sides okay so all the blue so all the blue arrows at the top those are not pumped that's just matched to a 50 ohm impedance exactly that's just dissipation exactly okay I saw a hand Mikko it was you yeah I was kind of curious about the reverse gain as function of the frequency offset you showed the forward cane to go to 25 dB but and the added noise you showed to be 1.3 but you didn't show the reverse gain does it behave nicely like you like you said that the reverse gain is about minus 25 dB at that sort of optimal point but it's yeah exactly kind of staying low when you go to offset in frequency yeah even get slower so okay maximum at the at the center it's maximum and then frequency gets slower so I had the plot actually but I took it out because of time but right okay maybe um I'll show it actually I'm also wondering like and you defined that to be the gain for the to the reverse direction if you put signal at that frequency and what's the power coming out at the other end at the same frequency right so that's kind of yeah but then how about the idlers etc like the danis was kind of curious that when you have like if you think about your tupan now you have to the reverse direction you have lots of noise power coming out at the very broad frequency band so are you are all of those frequencies do you have that isolation or is it only close to the close to that frequency point and then when you look at the idlers that those frequencies generate are those idlers also a low power no I think the idler is at high power I would say so because the idler also gets amplified as always in parametric amplification I think so the but this is I'm talking about the reverse direction that and the reverse direction that ah yeah I see yeah yeah also the idler gets suppressed they're both both both yeah because it's just the properties of the discrete functions so the yeah but in reverse direction idler and signal let's say you get suppressed and in the direct direction the left to right direction the gate amplified exponentially but noise signal idler but in different proportions so one can find a certain parameters where where the noise is still the noise the say the noise to signal ratio is it's good enough okay well thank you welcome okay thank you Thomas again thank you and thank you to all the the speaker of this morning session I think it's now time for lunch break long awaited lunch break I should say