 Now you are online. OK, now you hear me. OK, sorry. OK, so as I said, non-classical states are a resource for quantum information technology. And I will tell you about two different ways to create non-classical states that are described in these two papers. The first one is how to create, in a very simple way, a continuous stream of non-classical states in a rather different way than what people have done before. So it's a waveguide QED setting. So it's a qubit in a transmission line that you simply drive continuously. And I'll come into the details later on. The second story is towards continuous variable quantum computing. So it's a 3D cavity coupled to an ancillar qubit where we can make different complex non-classical states in a rather robust and high fidelity way. So I'll discuss that in the second half of the talk. Good, so the first description, the first story here is really the PhD thesis, a part of the PhD thesis of Jung Lu, who graduated almost a year ago. There are several people involved here. And this idea came out from two theory papers that were made at Chalmers. And then we have now implemented this. So that's the background. Maybe having some feedback there. I'll stand here. Good. So just a few words on existing single photon generators. So the simplest way to create a non-classical state is to make a single photon generator. And within circuit and waveguide QED, I think the first one was done by Andrew Hauke and Rob Sholkov's group in 2007, where they essentially coupled a qubit to a superconducting resonator. They sent in a pipe pulse. They excited the qubit and then let the qubit, the excitation decay into the cavity that was then released into a transmission line. That was later on refined by several groups, including the Ehtah group. So the second version here is the one you heard from Denis Vuillon earlier today. This is Cooper pair tunneling. So by the way, I should say that, of course, all these have different pros and cons. And one limitation of the first one here is that you're limited by the frequency of the cavity. So you're sort of locked to the cavity frequency. Here, if you tunnel a Cooper pair, you can then generate single photons. And you can do this in several different ways. And it can, in principle, be on demand, even if it's a bit more tricky than other on-demand sources. It's bright, and it can also do other things like send out entangled pairs and multiple photons. So a drawback with this one, as I see it, Denis will have to correct me if I'm wrong, but I don't think you can make a superposition between zero and one in this case, for instance, which you can in many of the other ones. So then single photon sources were also made in waveguide QED, where you essentially just have a qubit that is coupled to two channels. One is strongly coupled, and one is weakly coupled. And you send a very strong signal to do a pipe pulse in the weakly coupled, and then it decays out through the strongly coupled port and sends out the photon. This has several advantages. So it can, of course, be on demand, so you just pulse it. It can be frequency tunable, so you're no longer limited by the resonators as you are in these two cases. So the drawback here is maybe that there can be quite some leakage from of the coherent pulse that comes into excited qubit. So that sort of mixes with the single photon. So then in 2016, there were two suggestions for how to make other photon sources from Chalmers. And one of them, so this one here is to have a qubit in front of the end of a transmission line, and you can tune the distance between the atom and the end of the transmission line. And this was implemented in Chris Wilson's group at Waterloo. This is also on demand and it's frequency tunable. A nice thing here is that this also allows shaping of single photons, so you can tailor the envelope. You can also delay the release to also be able to separate the coherent pulse from the single photon pulse. And here, the one that we also implemented, this builds on having essentially a beam splitter here where you send a coherent signal to the atom and then what's reflected is a coherent signal plus a single photon, and then you cancel the coherent part through the second port of a directional coupler. That allows you to then essentially kill almost all of the coherent pulse so you get a very pure signal out. That's the advantage with this one, which can also then be on demand and frequency tunable. So these are all sources but they're all pulsed. So what I'm gonna show you now is a different kind of source, which is a continuous source. And this goes back to an experiment that we did when Chris Wilson was still at Chalmers in 2011 when we placed a single tube within an open transmission line and we measured the transmission and also the reflection. And we could see in the G2 that there was a clear anti-bunching of the reflected field. And then we thought, how can we make something a good device out of this? And it took us some time. But, and also can we measure the Vignor function of this that's coming back? So talking then with the theory people, they come up with these two papers here which has the following suggestion. So instead of having just an open transmission line, you have a half infinite transmission line. So you have a qubit up to a mirror. You also, and you send in the signal here, it bounces out here and you, we amplify it. And then we apply, so, but this is a continuous signal on resonance with the qubit. And then we apply a time filter here. And what they calculated is that if you adjust the amplitude here in a nice way, you can actually get a non-classical state out here, a stream of non-classical states. And the nice thing here is that you can, it's actually the, it's not the sender who decides when to collect the non-classical state, it's the receiver because it's the receiver who can sort of set his time filter whenever he wants. So compared to the previous thing that I showed, the modifications are that now we have a semi-infinite line, so a mirror. And also we draw, we adjust the drive amplitude to know the coherent reflection. I'll show you that on the next slide. And also we have this time filter that we can then also play with. And I'll show that too. So if you look at the reflection from a qubit in front of a mirror, it looks like this. This is plotting the real part and the imaginary part. And you can see that all of the action is in one quadrature, which here I call the real part. And if you go at really low power, then everything is coherent, reflected by the atom with this phase here. If you go at very high power, you saturate the qubit completely and it's reflected by the mirror, but with the opposite phase. So in the middle here, there's a point where the coherent reflection from the qubit and from the mirror are equal, but with opposite phase. So you kill the coherent reflection completely and everything that's reflected is incoherently reflected. And that is, it's in this incoherent reflection that you have the non-classical state. So what we do is we choose this amplitude, this drive amplitude here so that we kill the coherent reflection completely. Of course, if you think of it, coherent state is of course a classical state. So you want to kill that as much as possible because adding a coherent state to a non-classical state just makes it less non-classical. Then the question is, well, what kind of filter should you use to extract this because you have to define, since it's a continuous stream, you have to define the time mode of your state. And so we have checked two different types of filters, just a rectangular box car filter and a Gaussian filter. And then measuring the quadratures of what comes out and we use a Tupac from Lincoln Labs to do that, we can then construct the moments of the operators A and A dagger. So here are these and knowing these operators, you can then reconstruct the most likely Wigner function like this as described in this paper from an ETH. And so what do we get? I start with the box car filter and I gradually increase here the length of the filter. And you can see that here as you increase this length of the filter, you get more and more negative Wigner function here. And we can then, instead of plotting all the Wigner functions, we can extract the Wigner log negativity and plot that as a function of the box car length. And you can see here that at around two, so two here means two times T2. So it's normalized to T2. You have a maximum here of this state coming out. And you can see it's not quite a single photon, it's something else, but the amount of power in it is about the photon. Then we go to the Gaussian filter and you see a similar thing that you have also an increasing negativity of the Wigner function. And again, if you look at log negativity, there's also a maximum here around 0.5. And then you can wonder, well, you could also do a Lorentzian filter or something else, but you can show that the Gaussian filter is actually extremely close if not the optimal one. So this is a very different way to create a non-classical state than just to have a single photon source. So I'll try to point out the differences here. So, or the advantages with this. So this is a very simple construction. It's a qubit and it's a transmission line and you need a directional coupler. It's tunable in frequency. If you compare it to the pulse sources, it has very high efficiency. So essentially all the power that you send in is used in the non-classical state that comes out. If you pulse a qubit with the pi pulse, it typically a pi pulse contains of the order of 100 to 1000 photons and you sort of put in much more power than really needed. Also, since we put in so little power, there's no risk of populating the second excited state. So if you have a pulse source, there's a trade-off between having good properties and not populating the second state. We have a high rate because we don't need for the qubit to, we don't need to wait for the qubit to decay. So we just drive it continuously and we can pick out at every time slot we want, which is smaller than our time filter, we can pick out a resource, a non-classical state. And what I find is interesting and might open for new experiments is that it's the receiver that decides when the non-classical state comes, not the sender. So this opens opportunities. I don't have a brilliant idea on this, but I could imagine that if you want to do things like boson sampling with single photon sources, you could use these sources instead and do something in a more efficient way. So this is really a continuous source of Wigner negative states. Good, that was the first part. So I'll march on and talk about the second part. So the second part is really a part of Marina Kudras PhD thesis. And you can see there's a lot of people involved. And this is also done together with two companies into modulation products and quantum control. It's, this is not yet published, but it's on the archive. So just to make the motivation for why we want to do this. So we want to create in a 3D cavity, we want to create interesting states that you can encode information in. And of course, the idea here is that instead of making qubits and making logical qubits with the surface codes, et cetera, you could do error correction directly on the qubit itself before you go on and try to do some computation on this. And this field has of course been pioneered by the Yale group. So a few words on previous ways of making interesting states in these cavities. So the first one was really done by Max Hofheins when he was at UC Santa Barbara together with Andrew Cleland and John Martinez. Sorry. There they essentially put the qubit into resonance with the cavity and exchanged excitations with the cavity and bit by bit they built up an arbitrary state. That was very nice of course, but if you go to a very large state containing many excitations, that takes a long time. So then the Yale group did this kind of experiment where they use what Michel de Verre calls the Swiss army knife, using combinations of displacements and snap gates. I'll come back to explain what those are later on. But essentially you send in a coherent signal to the cavity and then you send in a number of signals to the qubit and that in that way you can actually create a non-classical state which is shown here. So a few years later they showed another method where they used full optimal control to tailor the two signals both to the cavity and to the qubit. And so that was sort of an alternative way of doing it and that's a bit faster. And then recently they have also come up with this where they use the so-called echo conditional displacement to do this and this was what they used to do the GKP state, the Gottesman Kitai of Preskill state. Now what I'm gonna show you here is that we have done something which is sort of a mixture of these two here. So we say that oh the displacement is so simple that we can do without optimizing and then we only optimize with optimal control the snap gate. That's in a few words what we're trying to do. So the system we have is a 3D cavity so it's one of these coaxial cavities with this post here and the electric field of this post oscillates then in this cavity and that gives a very high Q value and then you can stick in a chip here and we put in one made of silicon with a transplant qubit and a readout resonator. And here are the numbers for the qubit and the cavity. So you see the frequencies, the readout resonators at 7.2 gigahertz, the qubit as at 6.2 and then the cavity as at 4.5. And the qubit has decent coherence, 30 to 40 microseconds and the cavity is about 10 times more long-lived. So since a few years we've developed a method how to make really high Q cavities. So by doing, so you see if you just machine the cavity it has a big spread in the Q value from say 8 million to up to 80 million. But if you then first etch it and then anneal it you see this green crowd here you can get all of the cavities and we tested quite a few to all of them above 60 million. So that's sort of a reproducible way of really making good cavities. Of course, once you stick in your silicon chip it's gonna go down a bit but it's always good to start with a good cavity. So how do we do this then? Well, so we do it very similar to what the Yale group did first. We define a target state that we wanna do and then we optimize a sequence of displacements, snaps, displacements, snaps, displacements. And so first you choose how many snaps you wanna do and in our case it's two or three so it's not very many. But then you can optimize for these how big the displacements are so how big the alphas are and then how big the angles are on the different foc states I'll come back to that in a minute. So once you know these angles then you take this angle and then you put in the Hamiltonian parameters into a software that has been developed by this company Q-Control and that lets you calculate optimum microwave pulses the I and the Q, the two quadratures of the microwave pulse and we use then the hardware from intermodulation products to play up these signals. And in that way you can shorten the length of this snap gate from several microseconds to half a microsecond. So there's an improvement in time doing this. Then of course we apply the sequence and finally we measure the Wigner function. So this is done in the strongly dispersive regime so and that means that we can resolve the photon foc numbers. So if you do spectroscopy on the qubit and the cavity is empty you see just a single spectral line but if you fill up the cavity with in this case 1.8 photons then you are in fact see the different foc states zero, one, two, three, four. So each of these peaks corresponds to the cavity having a well-defined foc number. And the distance here, so this is resolved so this is three megahertz between these peaks here. And so normally what you would do in a snap is you would send in a signal on each of these so you would send in a frequency comb but you would have to have it long enough so that in frequency space it would be more narrow than the distance between those two and that limits the time of this. But so what we do instead is we do this optimal control and then we can do it faster. So just what are the displacement in the snaps? The displacement is rather obvious but you get some intuition if you look at the foc number so you start with vacuum. So this is the Wigner function and this is the probability of the different foc states. Then if you displace you send in a coherent state to the cavity of course then you displace the vacuum and you start to populate the cavity with the Poissonian distribution. And then you can displace it of course with a different phase and then you fill up more photons in the different foc numbers. And so what is a snap? So a snap is when you for a given foc state add a phase that you can control. And the way you do that is you go from the ground state of the qubit to the excited state with the pipe pulse and then you directly afterwards put in another pipe pulse but you rotate the phase and that means you're covering this blue area here which adds a berry phase to this foc number here. And since you can of course change the phase as you like you can then change this phase as you like. And so the way you do it here, first you make a displacement like this. Then you do this phase angle and then directly you get something that looks like a displaced foc state. So if this phase here is pi then you see that the population of the foc numbers has not changed. The only thing is that you have changed the phase of the zero of the vacuum. So an example how to do the single photon foc state here is you start by displacing. So you populate a certain number of the foc states here. You then do this phase control. You get this displaced foc state and then you displace it back by another amount and then you see that you have only the single photon foc state occupied. So it's a very simple way of creating a foc state. Now, so to summarize what we do is we have the cavity, we have the qubit and we have the readout. We do the sequence of displacements, snap displacements, snap and so on and then we stop with the displacement and then we have our state. So this is the state preparation and then we do the Wigner tomography which you do by first displacing and then doing a Ramsey on the qubit which is exactly one over two chi. And when you then measure, you get the value for this alpha in the Wigner function and then you just map out with different alphas over the complex plane and then you have the Wigner function. Of course, it takes quite a long time because you have to do this over the full plane but this is what we do and so now we come to the results. What did we achieve? So with two snap gates and three displacements with an overall time of 1.15 microseconds, we generated all these three states which all have an average number of two photons. The two photon foc state, a bi-nomial state, the superposition between zero and four and also Schrodinger's cat state alpha minus minus alpha. And you can see that we have decent fidelities here of the order of 94 to 99% fidelity. Now we also want to do more complex states and that we do here. So we can do the GKP state with three snaps and four displacements. And also we can do this cubic phase state which is thought to be a good resource for continuous variable quantum computing. I know it's a bit debated in the literature, how useful it is but let's see now we've made it. And to our knowledge, this is the first time anyone has made it. And so here the fidelities are a bit lower but the reason is of course that then we have to include more larger foc space. So for the GKP we actually go to the 17th foc state. Good and another advantage with this is that since you do this in sequence you can actually after each of these operations you can go in and do the Wigner function and check that it really happened what you thought it happened. So you can check then and of course you can solve the master equation and see what it should be in theory and you see that it agrees quite well on these steps. So we can assign fidelities to each of these steps then and see which one is sort of the difficult one. The last thing here is that we can also then show that this is a robust method. So we can vary different parameters. So the dispersive shift that we stick into the optimizer we can change the amplitude of displacement and we can also change the frequency and the amplitude of the snap gate. And you see here on this side here that we have done that and we have good agreement on how it varies with these parameters. So the red is the theoretical calculation and the black is the measurement. And you can then also compare that to what our simulations say if we would use a standard snap. And what you see is that our method is substantially more robust in terms of variations in the parameters. So to summarize, we've been able to do high fidelity Wigner negative states using snaps and displacements. We have optimized pulses to speed up the snap gates by four to eight times. And we've shown that these pulses are quite robust. And with this, I thank you for your attention and I also wanna say that we have post-doc positions announced at our homepage. Thank you very much. Thank you, Beth for this nice presentation. Time for some questions. Thank you for the nice talk. Maybe I'm rather confused. The photo numbers distribution is actually Poissonian. Is there any traces of a non Poissonian distribution? I guess I observed some... Yeah, so when you send in a coherent state, of course it's Poissonian, yeah. But once you have made these states, it's definitely non Poissonian. I understood that you said it was Poissonian, okay. Nice talk. I noticed that sometimes your experiment fidelity is better than the theory one. Yes. Is it because of the uncertainties? Yeah, so we have thought about that quite a lot. If I go back to the slide with that. So there's in two cases, so here you see our Fox state two is actually a bit better than our theory. And also here the DKP state is 1% better. So first of all, I should say that the uncertainty of our fidelity is about 1%. But it's also so you know that the estimate of how good it is is based on the T1 and T2. And as we know, T1 and T2 fluctuates. So what we think is happening is that in some cases we were lucky and that T1 was a bit longer than the average that we have measured. Because the numbers we put in is of course the average, but we know that there's quite a spread around that. So that's the explanation we have. I think that's the case, but I cannot be completely sure. Thank you very much for a very nice talk. I have a question about the first technique that you used. If you go to the Vigna, yeah, things like that, maybe the previous ones or whatever. They look, well of course there's some asymmetry, but even if you remove that in your head, they look like they're also displaced fox states. Do you know if the direction of the displacement has anything to do with the original coherent state that you use? We have not checked that, so I cannot say. So we, you know, this is of course not an ordinary fox state because for one thing, the time envelope of it is very different, but it also, there is this asymmetry that here you see that the negative part is below the center of the red part. And I don't really have an intuition what that means in terms, if you can identify that with some certain superposition of some states, but if you have some input, I'll be happy. So another last question here. Yeah, I'm wondering about actually two things. You said, you described that you completely canceled the coherent part, but is that, have you actually measured that that gives you then the highest WLN or could you still a little bit optimize the power to get a little bit more negativity? So here I should be clear that, so this, so you mean that this is, that we're right at this point. Yes, so we optimize that quite carefully because it's rather sensitive on that, so. Okay, and then this is the G2, how does the G2 look like for these states? Okay, so we have not measured G2 for these states, no. But they should be substantially better than 0.5 as we had before. But it should be substantially better because this qubit had much better coherence than the one we had 10 years ago. I have a question as well. Hatan Turechi. Yes, yes, this is Akhan. Thank you for taking my question. I was wondering this fidelity that was stated for theory. What does theory mean there and what does fidelity of theory mean? Yeah, it means that this optimization when you have your target state, that is the one we're trying to make. But even if you optimize the displacements and the angles, the state you get is not exactly the target state and it's fidelity of that state, but including the decoherence of the qubit that we say is the theory. Does that take into account the measurement chain or the amplifiers in the line? No, it doesn't, but the noise there is quite small. We have, so we have a dupa. And we average a lot and yeah, I think that's not the main contribution. I see. Okay, thank you. Okay, thank you. Let's rank per game.