 and I moved from Julie 26 years ago. Yeah, and actually I went there, I corrected it before the conference, but it's still there, so it's. We corrected it, it came back. Yeah, well, anyway. Just to make sure, I'm at the Cultural University Institute of Technology and that's been for already almost 13 years, 14 years. So, and I would like to present the work done by a very talented young PhD student, Jan Brehm, who graduated actually last year. And it will be about a race of superconducting qubits in the wave grid. So, the attitude to look at this is maybe, could be various, but we like to look at it as sort of kind of artificial medium made of something which behaves as a certain kind of material with particular properties for propagating electromagnetic waves. So, if one takes a material and eliminates with electromagnetic waves, then the limit that the distance between atoms is much smaller than the wavelengths, the medium behaves with its characteristic dielectric constant and magnetic permeability. And that's actually defines the penetration of light or waves through this material. And already more than 20 years ago, there was a kind of idea proposed by, mostly by electrical engineers, that one can actually design artificial medium made of out of identical small elements, much smaller than the wavelengths, which at certain frequency will have some peculiar properties. And one of those driving ideas was actually to implement negative epsilon mu, a particular frequency, which actually has a lot of interest at that time. However, if one goes this path, which was originally proposed in this paper, here, like using the splitter resonators, which were smaller than the wavelengths, but almost comparable, by introducing the superconductor one wins here frequency to nobility, and actually also very low loss. And we have done over past 10 years or so, experiments with classical superconducting circuits, like squids, which interact with electromagnetic waves, and which properties can be tuned by magnetic field. So you can tune the frequency by changing the Josephson inductance through the current, which is flowing in the ring. But most interesting, of course, is to use the capacity of superconducting circuits such that the circuits really behave as quantum systems. So now one can actually make a medium consisting of artificial atoms, which could be made much smaller than the wavelengths, and then they will show pristine quantum properties. And the first paper, which we had on that, was already almost a decade back. This is a paper by Pascal Maha in collaboration with the Institute of Photonic Technologies in Vienna, where we actually started the array of qubits in the waveguide. So the artificial atoms, which we like to take, are not just harmonic oscillators, as we have seen in this classical kind of prototypes, but really the elements, which have been talked about very often in this conference, these are transplant qubits, which are basically Josephson junctions shunted by capacitance. And these transplant qubits, they have slightly unharmonic potential. Therefore, one can couple by electromagnetic irradiation the ground state in the first excited state. And if the driving is not too hard, actually avoid excitation to high levels. So one can really work with them as real true-to-level systems. The device which we take is a transmon in this most kind of beloved configuration with this interdigitated capacitor. So you see here, this is the transmission line going in this, that was where the waves propagate. And this is the device, which consists of two large superconducting electrodes forming large capacitance. And then the transmon itself is a squid, which actually allows us to tune the Josephson energy of the device by changing the frequency of the transition between the ground state and the first excited state in the range between three and to eight gigahertz. And as you have heard already in this meeting that these devices are weakly unharmonic. So the harmonicity in our particular devices is in the range of 300 megahertz. And it's actually, of course, negative because of the soft potential, which we have seen in the previous slide. So as I mentioned, the number of papers, I forgot to update this slide to put some reference here, but the minds easy to find. So basically, there would be a number of experiments of arrays of superconducting qubits placed up in the cavity. And in this case, one can show Davis Cummings Hamiltonian behavior. One can show collective coupling and coupling to the single mode of kind of collective excitation in the system. In my talk today, I'm going to speak about arrays really placed in a one dimensional space in transmission line or one dimensional waveguide. And here you see a picture of our sample holder. So basically the whole device is quite small. It's like five to 10 or five to eight millimeter device which is placed in a microwave sample holder where we could measure the transmission or reflection from a circuit at microwave frequencies. Now, if you take arrays of qubits, arrays of qubits, there have been a number of theoretical works predicting various interesting effects. And between them, just to name a few were super and subradians. There was an old proposal already before first experiments started about band gap engineering. That's something I'm going to touch today. One can use a device for quantum memory. And that's what I'm also going to go into mention today to generate a non-classical light that's something which is still kind of open field to what many experiments done there. One can do lazing, one can do single photon detection and so forth. So the major experimental challenge here is really to make the artificial atoms as nice as natural atoms. And the difficulty is that you can never make them identical. So then of course by technology, they will be slightly different. And that means there will be some spread of frequencies of these artificial atoms, which will really make them being distinctly different from their natural prototypes. So considering an array of qubits placed in the wave guide, one can write this Hamiltonian where you see the Hamiltonian of qubits summed over all the qubits. Here, one can see here the waves propagating in the wave guide. This is this term. And then there is a coupling term, a type of like coupling between those propagating waves and qubits. And they have been shown by several theoretical groups that it's kind of instructive to integrate out photonic degrees of freedom. And then the effective Hamiltonian becomes the qubit Hamiltonian consisting of these two terms. And then there will be infinite range interaction between qubits given by the last term. Now, this infinite range interaction mediated by photons comes in two flavors. First is the flavor. It has a complex character and imaginary and real part. And one flavor is a collective decay, which depends on the cosine of a distance between the qubits. So this is the gamma ij. And the second is the exchange interaction, which is proportional to the sine. So basically, by changing the distance between the qubits, one can modify those two types of features. And the decay, of course, means that we'll be enhanced the probability of losing the photons in the wave guide due to the collective interaction between the qubits and the propagating photons. So when one takes the single photon manifold and diagonalizes the Hamiltonian, then one gets eigenmodes for this Hamiltonian. And there are so-called superident modes, which are characterized by enhanced decay, so that basically the imaginary part of those eigenmodes is larger than the decay of individual qubits. So it's the decay quicker. And there are also sub-radiant modes, which actually don't want to decay, but of course, in this case, the decay characteristic is limited by the intrinsic loss, basically. The loss, which is not the radiation in the wave guide, but rather some intrinsic loss in the qubits themselves. And one can see that actually on this plot, the color shows the imaginary part of those modes, and the vertical axis shows the real part. So if one increase the number of qubits, then there is a formation of a band gap between the super-radiant mode, which is the upper one, and the variety of the sub-radiant modes, which are the low ones, which in the limit of very large number of qubits becomes really a gap. And there are these polyurethonic branches, one limited by the upper range of the gap and the other one is limited by the lower boundary of the gap. And here in the gap, basically the radiation from the array of qubits is being reflected back in the wave guide. So all these modes actually are characterized by these polyurethonic type excitations of these artificial atoms interacting with one-dimensional electromagnetic waves propagating in the wave guide. So just without really thinking much, one can place a lot of qubits on the chip and then do something with them. And that's what we have done immediately. We thought, but we should perhaps see something. So Jan made a great effort of making really nearly identical qubits here, at least to the perfection he could do. But still there was, of course, spread of frequencies in this large array. So here you see 90 qubits placed in one-dimensional wave guide. We apply microwave from here and we take the microwave out. Basically, we measure the transmission. And in this particular case, we coupled, in this first experiment, we coupled the qubits very strongly to the wave guide. So the decay was very large. So the T1 time was limited basically by this radiation in one-dimensional wave guide and T2 was T1 limited, of course, in this case. So basically just trying to look if we could see signature of these qubits acting together. And the prediction from the model was that we should have a band gap at some frequency here. But of course, if you take the spread of frequencies of qubits, then we get these great curves. This is some randomly chosen configuration of disorder in the array, which we just wanted to have a look in this simulation. And the experiment shows indeed a band gap, which we see here at about five gigahertz. So this is the band gap. The blue range is the way the transmission is very low. Basically, the waves get reflected from the array. And the signature of two-level behavior is that there is a saturation here at a particular power. And if you can take some of the lines which we see there as single qubits, which are often frequency from this large array of qubits, then the saturation for them was around minus 120 dBm. Whereas the saturation for the larger array was actually at a power about 18 to 20 qubits higher. So from that, we could estimate that actually this band gap, which of course has some spread of frequencies inside it, there is about 60, 70 qubits participating. But as we cannot control really the, we cannot make qubits identical. We cannot control their frequencies. So of course, it's not really much what we could do with that. You could sort of have a look and get an idea, but you cannot really do some fine tuning and clever experiments with that. So this actually brought us to the point that we decided to make the whole array tunable. Basically, we need to change the frequencies of all the artificial atoms. And doing this in experiment requires, of course, eight controls. Or if you want to go more, it's possible, but it really becomes a really bulky experiment. So we decided just to take eight qubits. And this is our sample here. Again, similar geometry as before. We have chosen the distance between qubits to be of the order of 400 microns, which actually corresponds to the interqubit distance at the frequency of the resonances being smaller than the wavelengths. So we are in the metamaterial limit, such that actually the qubits are pretty close together, but of course they occupy certain fraction of the wavelengths in the waveguide. And as you see here, all the qubits, in contrast to the previous picture, they have now this control line which you see here. So basically we could run the current through each qubit, I have a problem with this pointer. We run the current through the each qubit here through this line, okay? And this changes the flux in this loop. So basically we could individually control the frequency of every qubit in this eight qubit array. So first thing to do is to characterize individual qubits. So in here we tune seven qubits away, put at the frequency of our interest, which was around eight gigahertz, 7.88 gigahertz, a single qubit, and then measure the transmission through this array. We measure this coefficient S to one. And as was observed in pioneering work of Oleg Astafyev that time at NEC and followed also by other studies in particular Chalmers, this array shows, the single qubit shows resonant fluorescence. Basically if a photon propagates in a waveguide and this photon is resonant with the transition between the ground state and the first excited state, the qubit gets excited and then remits the photon back into the waveguide with opposite phase. So basically the photon gets 100% reflection probability from the qubit. And in this case, this is characterized by the figure of merit, which is called extinction. So basically it's a ratio of this full transmission. It's actually a ratio of the full transmission to the transmission at the qubit frequency. And you see in our case it's really high, it's about 99%. So again, the qubit was very strongly coupled. The reason is the qubit is very strongly coupled to one dimensional waveguide. It's coherence times T1 is radiation limited is about 25 nanoseconds. And the corresponding T2 lag time, which is the defacing is estimated from this circle feed is about twice larger. So this is the thing which has been already shown for individual qubits. So let's see what taking more qubits brings to that. So now what we do, we tune one by one qubits to the resonance of the first qubit which is already sitting there. So when we put two qubits at the resonance and the other six qubits are away, we see that actually the lines modifies such that we observe this quite non-monotonic, actually not symmetric, and you see this little peak which is rising on the left of the main minimum. If you bring more qubits to the resonance, then you see this peak becomes quite clear. It's a pretty sharp feature. And then if you bring four, it really grows up. And if you bring all eight qubits, then what you observe is that actually there is a wide gap here on the right. And then there is this peak and then also the appearance of precursor of the additional peak which actually the dark states. So the appearance of this band gap is actually a signature of the bright state, but on the side of the bright state you see also the appearance of these dark states. And it's a bit more convenient to measure this, not in transmission, but in reflection. And that's what next picture shows. So this is now eight qubits measured in transmission and this is their measurements in reflection. So you see this sharp feature which is this dark state. One of the dark states appears here as a dip. And from that we could actually estimate quite well the decay rate or the coherence introduced by the complex part of the eigenvalues. And what we see is that the resonant ensemble of qubits showed the increase of the corresponding decay rate as shown on this picture. So this is the brightest of the dark states and the theory which is described also in this paper and also in other papers published earlier predicts the scaling at the power and to the free for the brightest dark state. And this is interesting observation which we observe here in experiment. These are the crosses here and triangles is the numerics and the dashed line is actually the analytical treatment done by Aleksandr Padubny about this problem. So basically here this is the other next dark state next to it. And the same type of approach actually predicts the universal scaling for the darkest of the dark states which should go as n in the power minus three. So I'm not discussing these details here actually. We don't have enough data to confirm that because we cannot really observe this darkest subradian state for the reason that we are limited by the intrinsic decoherence of individual qubits themselves and not only by ray of decay. So we cannot really make measurements of the line which is smaller than the natural line bits of our qubits which was corresponding to the characteristic to one time of the order of few microseconds. So this plot shows again the dependence of the transmission. This is S12 by bringing qubits one by one into the resonance. And then you see here this dark area here is this band gap which appears in the transmission spectrum. And the crosses are the corresponding predicted eigenvalues for the brightest mode. For this is the superradian mode and this are the subradian modes on the left. So for our particular array the width of this band gap is about twice the width of the individual qubit line limited by the radiation. And for infinity long array the theory predicts the increase of up to 6.3. This is for particular chosen distance between the qubits which of course is limited by the size of our sample here. So the next thing we could do we could actually do many interesting configurations of frequencies of qubits and measure the transmission or reflection. This is a data measured in reflection again. You see the experiment and calculation on the right experimenters on the left. So what we do here we fix the frequency of seven qubits at this horizontal black line and then we move the frequency of the qubit of the eight qubit through this frequency. So we move it along this dashed line and then as we move we measure the reflection coefficient s to two and you see that the minimum in reflection basically this is the appearance of the transmission is shown by this dark area, dark blue area. We could actually see that the whole number of lines depends on the detuning of this individual qubit relative to the rest of the qubits in the array. And basically one can explain these features by condotype behavior where the qubit sort of dresses the state of the array or other way around the array dresses the state of the qubit. So basically the interaction here is very strongly dependent on this detuning but we have really nice agreement between theory and experiment in this kind of example here. Next thing we could of course, this is all single photon manifold. So all that can be explained also by classical equations of motion but we are dealing really with two level systems and the first signature of quantumness which we could have here is that we could increase the power and then see the saturation of the band gap of the gap of the transmission deep which is introduced by one qubit. And then when we take eight qubits on resonance we see that actually not only single qubit saturates but also the array of eight qubit saturates and basically the band gap disappears at the power which is significantly higher than the power of individual qubits and here you see the comparison of the transmission coefficient as a function of power. So you see the saturation comes at lower power for single qubit and then for eight qubit it's actually significant to larger. So this is the first kind of signature that we deal here with two level systems. Second quantum behavior which is well known in atomic physics is the observation of the otolotone splitting and that's what you could do by taking atoms or any kind of multi-level systems. So if you have three levels in your quantum system then you could drive the transition between the first excited and the second excited state and then you can measure the intensity of the probe tone which is shown here. This is the transition between zero and one as a function of the amplitude or power of the driving tone applied to the transition one too. This is described in the rotating frame by this Hamiltonian here and basically what it brings the driving tone dresses the first excited state with these two states with the splitting between them omega C which is proportional to the driving field amplitude and here you see the in our example when we don't derive our system we see a deepened transmission. This is shown here so this low transmission is on the left high transmission is on the right. So this is our band gap and when we apply now the control tone we basically see two peaks. We see the splitting of this first excited state of the ensemble and here we could actually do it as a function of power and this is shown here. So we observe this the dependence of the transmission as you see this is the band gap and then the band gap splits and then in the middle of the band gap we see the transmission window which is controlled by by the applied microwave power. So the difference is omega C the distance between them in frequency in between the address states is actually shown by the distance between this two red dashed lines. And in theory what we observe here is pretty similar. So this is invited question what are those things? Let me tell it right away. This is basically the we believe this is a signature that our qubits not identical. So we could tune the transition between zero and one to the same frequency but it doesn't mean that we could make qubits ideally identical in their harmonicity. So we'll have slightly different transition between one and two. And that's in a reason for having this kind of multiplets which you usually observe here. So this was done with eight qubits and then if we reduce number of qubits actually we could take this multiplets away. Now when we do the auto-transpleting this is now a picture of the dispersion in the race. So we see that we have this very same polyurethonic branches, the lower branch and the upper branch. But here in the middle between the two gaps which we obtain on the driving tone we have the dispersion which can be seen in this middle part of this figure and this dispersion actually corresponds to pretty flat band here. So this flat band is characterized by this group velocity which is the omega over decay which is actually extremely low for the left picture. Increases if you go from left to the right but this is a kind of feature which has been also used in experiments with atoms to demonstrate slow light. So what we decided to do then we decided to play with this a little bit and what we did, we took now seven qubits just to avoid this extra multiplet which you see now the picture is more uniform. It's very same picture of the otlatone splitting which we already seen in the previous slides. And then if you do the cut through this various colored lines so my point is okay it works. These are three lines and these are three cuts corresponding to these lines. From these cuts we could measure the real and imaginary part of the reflection from the array and that's what, excuse me, transmission from the array and then we actually observe from the slope of the complex transmission. We can observe from the face of an argument. We can observe actually calculate the group velocity. So it's actually inversely proportional to the derivative of the plot which is shown here by this dashed line. So from that we can actually estimate how latch is the change of the velocity of propagating electromagnetic waves and this actually is done on this plot so this data now taken a different power. The crosses, the blue crosses are experimental points and then the numerics gives the orange line and then there are two limits in the theoretical calculation which is given by this formulas below done again by Sasha Padubny showing the behavior of a low power and for high power. And basically the pulse delay or the group index which we get here is as large as almost 2000. So we can go above 1500 we slow down the electromagnetic waves by more than 10,000 here. So this is obtained from the spectrum so this looks interesting but maybe not enough convincing for experimentalists and we are such experimentalists that we really wanted to make sure that we have it as we think. So we decided also to make the pulse experiment. So we were then forming a pulse of the duration about 50 nanoseconds and then sending the pulse and measuring the true delay of this pulse. And here you see what we have done in our system. So this is the black pulse is the initial pulse which we have done with all the qubits tuned away. So this is the conventional pulse propagation. And then on the right you see the pulses which were obtained at different powers. Now one has to pay attention here that we cannot really go very close here to the slowest light because our pulse has certain width in time. And if you do the Fourier transform it means we also cover certain range of frequency. So if you go to very small powers where the phase delay is the largest then at that point we will have this kind of array working as a filter. So we will not get all the pulse propagating through. But if we do it in a reasonable kind of compromise between the widths of this Gaussian pulse in time and the stiffness of the group velocity in the region of the enhanced transmission basically we could have these plots which you see here on the right. And then the summary which I've already shown previously is fully confirmed. Now again on this now summary plot you see that we had the crosses. This is the group index determined or pulse delay determined from the spectroscopy where we measured the phase of the transmitted signal and calculated the derivative spectrum and the orange triangles are essentially directly measured pulse delays. And with directly measured pulses we see delays of almost 1,500. So the group index up to 1,500 in time result experiments. So this is really a very nice agreement. And basically this invites to use this arrays for example, quantum memory. Though of course one for quantum memory one really wants to have highly coherent qubits and these qubits are official atoms. And these qubits are not more coherent than the qubits which we otherwise have if they're not really limited by by Purcell or similar effects in their coherence. So with that I would like to conclude. So here you see this picture of Jan who really done all these experiments very systematically from fabricating samples to doing all the measurements and doing the numerical calculations. And his experiments have demonstrated fully controllable eight qubit material in the waveguide. We observed the tunable and saturated paleritonic band gap. Observed also the scaling of the decay of subredient and subredent modes. And interesting enough we also find here the capability of this one dimensional array of qubits to slow down electromagnetic waves and show a variety of quantum effects. With that I would like to thank you for your attention. Thank you for the nice talk. We have time for questions. Very nice work I think. How scalable is that? And is there a way to control the anharmonicity so that you can avoid the problem of the multiple, the multiplets? Well I guess for controlling a harmonicity one would need to come with more controls for each qubit. And this would really complicate the whole thing. And scalability once we need to tune the frequency. Well we have done also experiment with tunable qubits up to 25 qubits. It's a different experiment with the cavity which I didn't speak about today. So I think 25 maybe 50 is doable. But on the other hand we also see what is going on in this field. So now there are experimental tools to tune quantum circuits with many qubits with complexity to roughly 100. So of course this capability also extends with the growing up know-how in the field. So I think there will be fundamental interesting questions what you could do with let's say 100 qubits. I think it's doable. But of course one has to have very good reason to do it because it really complicates the whole setup. Yeah. Thank you. Have you thought about going to metamaterials with some kind of typological properties where you would be kind of more immune to these small imperfections which are inherent to like the nanofabrication? Are you meant to make qubits more identical? No, no, to let's say to have some kind of, I don't know, like you have a chain for you to go to some kind of typological phase of your chain where in which you have, you would have some gap but which would be connected to the typology of the chain. For example, you could have some SSH chain, for example, in which the properties of the chain would be more immune to fluctuations of the individual component of the chain. Well, actually, we have done exactly that with non-tunable qubits. We have done this SSH quantum simulator with 11 qubits and it works in spite of this order but this was without tunable qubits. So this was basically using 11 qubits with alternating coupling. And then we were able, that's a different paper which I didn't spec about today, we were able to observe also localized edge states not only for single photons but also for doublons. So there is a robustness of this type of systems, a certain degree but really if you want to fine tune and resolve the narrow line duct modes, I think there is no way that you could really do it with array with spread which is typically, even if you do all the best for transmodes, I think it's hard to get frequencies below, let's say, 3% spread. This is really difficult. Yeah. Okay, thank you. Thank you very much Aleksei for the nice presentation. I understand that the goal is to make these qubits more identical but on the other hand, there is interesting physics when they are not identical. Yes. So for example, I mean in the extreme case where many of the arrays of these which you may have Anderson localization. So is there something that you, where you see signatures of these effects? Yes, exactly. So exactly with the same experiment we have a paper collaboration with Sasha Mirlin group in Kalsa where we looked at the effect of disorder. Actually, we could do a variety of things here. I didn't have time to speak about. We could make, for example, the dispersion engineering by making the two clusters of qubits and then we get a very similar steepness of the group velocity which makes it possible to make slow light not by auto tones but by just dispersion engineering. And we basically were able to observe signatures which was similar to localization by introducing different degrees of disorder. And this paper actually just appeared in physical review A. I think it's end of last year. Yeah. Yeah. Thank you. Very nice talk. I'm very glad to see that. Now we have more and more control on many more qubits. I would have a question about the quantum net. I believe it was slide 19 when you look at this poor dependence. Did you, because if I am correct, this sub-radiant state is at 7 on the right plot is at 7.89 gigahertz, right? Yes. So what would happen there? Because the orange line is like in a radian state, but what would happen for the sub-radiant state as a poor dependence? And do you expect something very peculiar there? Maybe I didn't get your question quite. So if you look at the poor dependence at 7.89, yeah. What do you, there? This is the feature corresponding to one of the sub-radiant states. Yes, exactly. And what is there any, I mean, what would be the poor dependence of that? Or what would be the quantum as you said of this sub-radiant state? Well, we didn't really study that. I think we have also not done systematic numerical simulations for that. But it's interesting why the state saturates earlier than the other state. I believe if you, I mean, intuitively, if you have the sub-radiant state, that means that actually the cube is not oscillating in phase. So with the same kind of field which it drives them, they tend to do different type of motion. So they basically not do it like this, but they start doing like this. And how they drive sort of less promote, you less promote the state to exist. So I think it's more fragile and the situation of this sub-radiant state should naturally would occur before the situation of the super-radiant state. This is an intuitive picture, but we have not done any calculations on that. Yeah, and then, because it's non-monotonic obviously here. And so you explain the why, the first why it should decrease. And then the fact that it goes up again is just that you kill all the cubits. Or, I mean, again, if we speak about waving physics. I think this is where you basically have still robust survival of the bright state. And this is where the dark state is sort of eaten up by the bright because you drive them very hard. The asymmetry that you observe this slope here is due to the not ideal Lorentzian, which we also have for a single cubit. And that's a feature that's not really ideally coupled, so to say, in a symmetric way. It has a kind of condor type resonance shape. Okay, good, thank you. Other question? If not, we thank the speaker. And all the speakers, I mean. Thank you.