 who's coming from Argentina from Instituto Baiseiro and the talk is about dissipative entanglement generation between superconducting qubits coupled to a resonator and driven by microwave fields. So please, Maria. Okay, could you hear me? Okay, it's a pleasure to be here. I would like to thank the organizers, especially Cyprian and, well, Christine for all the efforts and of course the local organizers and New Yorking. It's a real pleasure to be here and it's my first in-person talk after the pandemic. So I'm very happy to give it here. So I have something. Okay. So what I'm going to tell you today is I'm going to present a new protocol to generate entanglement in circuit-quady architectures. Mainly, I'm going to consider two qubits which are coupled to a common mode of a resonator and what we are going to use is two key ingredients. We are going to drive the qubits through the energy landscape and to use dissipation mainly due to photon losses in the cavity in the resonator in order to stabilize an entangled state, a ball state. So this is a work that has been done in collaboration mainly with my students, and my longstanding collaborator, Daniel Dominguez. And along the years, we have been benefitted from the collaboration with former PhD students and also with Leandro Tossi, who is in charge of the experiments that we are going to carry on on these devices. So as I said, I'm going to start with... Well, I don't know what's going on. Sorry about that. I'm going to start with a brief introduction to superconducting qubits on circuit-quady. I don't need to give so many... I mean, introduction here to this audience. Then I'm going to mention some previous dissipative entanglement generation protocols tested previously in circuit-quady architectures. And then I'm going to give you a brief introduction to Landau-Sennarys-Dukever interferometry, which is the driving protocol that we are going to use to generate entanglement. And at the end, I'm going to follow... I mean, to focus on the system under study, which is, I mean, as I said, two driven qubits coupled to a resonator that experience photon losses. And I present the protocol and the summary at the end. So I don't need to say that superconducting qubits are essentially no-linear, I mean, quantum resonators in which the two lowest levels conforms the qubit, okay? The no-linearity comes from just some chance with behaves and no-linear inductance. And, of course, being a two-level system, there is a general Hamiltonian to describe these superconducting qubits in terms of, I mean, Pauli matrices. And in this case, what is called the detuning is an energy scale that will be externally controlled depending on the type of superconducting qubits that we are taking into consideration. And the other energy scale is the gap, which is the energy separation between the two levels of the qubit at zero detuning. For example, in the case of the flux qubit, the detuning is proportionally to the difference between the external applied flux to the device and the half-flux quantum, and it's also proportionally to the supercurrent in such a way that if this external flux is lower or higher than the half-flux quantum, we can tune, for example, in a flux qubit, the clockwise or anti-clockwise supercurrents that consist on the two states of the superconducting computing basis. So, well, circuit quity, I mean, also I don't need to give some details, but for those of you who are not in the subject, I am essentially talking about resonators, planar resonators that confine one or several modes of quantized electromagnetic field. Typically, this is a central conductor, but it's sandwiched between two rounded planes, and the device can be designed in order to contain half over two modes of the electromagnetic field, and the typical operating frequencies are between five and 10 gigahertz. So, what we are going to consider in these architectures is a system of two coupled qubits. The qubits are, I mean, indeed, not coupled directly, but are coupled to the same mode of the resonator. And this is, I mean, essentially the typical architecture, I originally proposed in this preliminary, this former work of the War rough group in which we have the qubits, the two qubits coupled to a transmission line resonator. And also, in a long years, these architectures have been employed in several experiments. I have this example, for example, here, which is a recent experiment in the group of Go. So, I'm focused on entanglement generation. What is entanglement generation? Well, it's trying to generate and manipulate two qubit entangled states of this type, the typical bell states that everyone knows between two qubits, okay? So, up to now, most of the strategies to generate entanglement had been based on unitary evolutions, adiabatic quantum computation or even gate quantum computation, applying two qubit gates in order to generate, for example, entangled states. But some years ago, people realized that, I mean, these type of protocols rely on unitary evolutions. And qubits are, I mean, noise is always there, so people think that a way in which could be profit from noise is to engineer noise in a way that noise can be used to stabilize an entangled state. This had been early proposals in the field of quantum optics by people like Syrac, Vestrat, or Soler. But the first implementations in circuit quid are at least in the group of Michel de Borre and Yale and in the group of Sidiki in Princeton. In both cases, they are considering circuit quid architectures. In the case of the Yale group, they are, I mean, two qubits coupled to a 3D resonator. And the resonator is continuously driven in a, I mean, resonant way. And they applied a sort of five drive-ins to the system in order to generate a highly, I mean, excited state. And then by tuning the decays rate, they can stabilize an entangled state. There was the first proposal, this one, the first proposal to generate a sort of autonomous entanglement stabilization without needed special feedback on the system. And this other proposal is also in circuit quid, but now they have a two qubit system coupled to both resonators. So they tune the couplings of the resonator, the asymmetry of the coupling of the resonators in order that, depending on the asymmetries, these are the gammas that, in a way, give the coupling to the resonators. They can stabilize even a singlet or a triplet state. But what I'm going to emphasize here is that both protocols rely on driving the cavity, okay, weak and resonantly, and tuning the relaxation rates in order to stabilize an entangled state. What we are going to propose is so different protocols. We are not going to drive the cavity. We are going to drive the qubits with a microwave signal beyond the resonant and weak driving regime that was, I mean, in generally used in the former approaches. So what we are going to do is to do what is called the Landau-Sennacher-Steukel driving protocols in where the system can be drive through the energy landscape with strong signals and out-of-resonance signals, okay? There will be signals that are modulated in amplitude and instead of being modulated, in a way, in frequency. And we are going to use dissipation by cavity photon losses in order to stabilize a two-qubit entangled state. So a brief flash on, I mean, what is Landau-Sennacher-Steukel's interferometry? Landau-Sennacher, well, it's a well-known protocol when you have a two-level system, that depends on an external parameter that varies on time. If this parameter depends linearly on time, I mean, the parameter has a constant velocity. The Landau-Sennacher transition probability between the low and high excited state is essentially an exponential on this parameter, which is delta, which is the ratio between the square of energy gaps over, essentially, the velocity of the driving, okay? So in the diabetic limit, there was no transitions, for example. But if you have an harmonic signal, that's the one that we are considering here, we have to take into account that the avoided crossing acts as a sort of beam splitter for the occupation's probabilities. Because as you apply an external signal, for example, here, with some given amplitude, you have a first passage to the avoided crossings in which the occupation, I mean, the transmitted and reflected population interfered again at the second passage and the avoided crossing. So dependence on the phase is accumulated. You can have constructive or destructive interference. So in the regime that we are exploring now, it's what is called the fast driving regime, in which the velocity of the, essentially the amplitude times the frequency of the signal is much larger than the square of the gap. The constructive interference can be shown to be this equation here. These phases can be computed. These phases are the accumulated phases between the first and second passage through the avoided crossings. And the other is the phase between the second passage and the end of the protocol. So these phases can be computed. And the resonance condition is essentially, okay, it gives you when the difference, energy difference between the two levels, which is, I mean, mainly epsilon naught in this case, it's an integer multiple of the frequency. These are the resonance condition from Landau-Senners-Dukevler interference. So at this resonant condition, you are going to have the maximum transfer of population. So the interferometric patterns can be shown in a 2D plot. As a function of the tuning and the amplitude of the driving. And you see the resonances and integer values of the external detuning, the DC detuning. But what is interesting is that these resonances are modulated by vessel functions. So if you set on resonances, you see that you have modulation and you have nodes. You have points in which the transition probability is suppressed. These points are the zeros of the vessel function. You have a fingerprint of this type of interferometry. And this can, I mean, gives you some well-known phenomena, like, for example, co-hidden destruction of tunneling at these points. So this has been, I mean, this is our simulation, and this is the experiment of the group of MIT, of the group of Will Oliver. So they propose this kind of a heterocopy to test the energy spectrum of the device for the shots on flux qubits many years ago. I mean, they propose this kind of interferometry, as I said. If they locate the position of these resonances by adjusting the transition with a Landau Center probability, they can extract, which is the gap of the energy spectrum and the position of the avoided crossing. So they can reconstruct the energy spectrum of the device from the shots of some flux qubit from this kind of experiment. And then they can do all these amplitude asbestoscopy because instead of changing the frequency, they have a fixed frequency, a non-resonant frequency, and they modulate the signal with the amplitude, which in this case is this external voltage. So neither resonant nor weak-riving are necessary to induce the transitions, and this is one of the main message that I want to give you. So, as I said, we are going to study a system that has a single mode of quantum resonator. So the Hamiltonian is this one. We have the Hamiltonian of the two qubits. We are going to drive the qubits through this, I mean, epsilon temporal dependence with the driving. It's what we, an harmonic driving. And the qubits are coupled with, I mean, slightly, and so I'm going to stress in the next slides. The couplings could be, I mean, more or less the same, but it's slightly different. I mean, could be some experimental, I mean, difference that it's going to be enough to, for the protocol that we are going to consider. But as I said, the resonator is coupled to the environment. We are going to consider a thermal bus, okay? So the full Hamiltonian of the system complete is this one that I'll show you here, the Hamiltonian of the bus, and some coupling Hamiltonian between the bus and the environment, okay? The usual way in which we describe the bus is the bus of anomic spectral density, okay? And we are going to consider that, as I said, the resonator is coupled to the bus through this operator that essentially is, gives the photon losses in the resonator because this does not conserve the number of excitations. So this is a sort of more technical slide. I'm going only to say that the strategy that you use to solve the open system dynamics is through the Floquet-Bormarkov equation. The Floquet-Bormarkov equation is an extension of the well-known Bormarkov equation to deal with time-dependent system that depends harmonically on time, okay? So instead of writing the equation in the standard, I mean, instantaneous basis, we use the Floquet basis, which is the natural basis to study time-dependent periodic systems. So with this, we can follow the dynamics of the reduced density matrix of the system and also we can study the stationary state of the system. That's, I mean, once we have the population on the stationary state, we are going to compute the quantities that we need. So first of all, as I said, we are driving the system, the two-qubit system through the energy spectrum, okay, with a Landau-Senners-Jukebert protocol. So the first thing that I want to mention is, I mean, which is the structure of the energy spectrum. Away from the body crossings, I mean, we have this structure in which we have states with our essentially product states of the two qubits times the resonator. And we have a state, our entangled state with a given number of photons. These states are maximal entangled states that essentially away from the body crossings that I am showing here in the spectrum are not sensitive to the external detuning in this case. So the idea is to identify what are the avoided crossings that can be reached when we put a driving on the system. So we put a driving, okay, we drive both qubits with the same protocol, okay? We set on this zero detuning and we drive with this signal both qubits. And as you see, as we fix the amplitude of the driving, we can reach these avoided crossings that are here. Each of these avoided crossings are essentially a complicated bunch of levels that interact, but we can identify two different types of avoided crossings. We have large gaps which are these circles here in which these states are mixed, and they are mixed with an energy scale of the order of the number, I mean, goes to the square root of the number of photons in the resonator times the coupling to the resonator. And I say both coupling are more or less the same, so the energy scale is this one, but it's a large energy scale for these gaps. But we have also smaller gaps, and these smaller gaps essentially mix this entangled C minus state with n photons with a linear combination of these two states. And these smaller energy gaps are of the order of energy delta G, which is delta G, the difference in both detunions. So we have these two energy scales, and this is interesting because, as we have these two energy scales, we are going to have different Landau-Zener type transitions through these avoided crossings. So the protocol is essentially the following. We are going to drive the system, starting from this initial state. We are applying this signal. We are going to excite the system from a given initial state which is a separable state. We are going to populate this entangled state with one photon, and then by the mechanisms of photon loss, we are going to populate the zero photons C minus state, a fully entangled two qubit state. But what is relevant for this protocol is that, as I am going to show you, this state is not connected by Landau-Zener transitions to other states because if you manage to populate this state, this state is going to be repopulated or depopulated by the driving the protocol fails. So here I'm showing you the interferometric patterns for the two relevant transitions that we are going to consider. As I say, we drive the system from the initial state to the state which is sine minus with one photon, and this is the landscape of the resonance. We observe resonances and integer numbers of the frequency of the cavity. Now the frequency of the cavity, as I showed you, is the energy scale that separates this transition because it is the separation between the flat levels. So now the energy condition for the Landau-Zener-Stukelberg resonances is this one, and the transitions has a width that is consistent with, I showed you before, the resonances are modulated by these vessel functions, but now the energy scale is not the delta that I showed you before, it's the delta G which is the difference, the energy scale in the gaps. This is essentially an analogy for two-level Landau-Zener transition. But what is interesting here that a curvature of the resonances of the order of this quantity, G1 times G2 over omega, is evident. And this curvature does not appear for two-level system. In two-level system you have the spectroscopy, you have the flat resonances, at least modulated by the vessel functions, but you don't have this curvature. And well, we have shown that this curvature is due because we have a complicated transition in which four levels are involved, and we can show explicitly that the curvature arrives for the two-level complicated avoided crossing that we have in the gap. But what is interesting here is that along these resonances you have the maximum transfer of population from the initial state to the state one sign minus. But on the other hand, what I'm showing you here on the right is the transition from the zero sign minus, which is the state that I'm going to read, to other states. And what's interesting is that we can select wide regions in parameter space in which this transition is on resonances but this transition is out of resonances. So if we play with the external parameters, we can populate this state, for example, in this region, and we stabilize the attained state because this state cannot transitionate to other states. So this is essentially the trick that we are going to use to select region in the plane amplitude and frequency of the resonator, where a unitary resonances to the state one sign minus is stimulated but no unitary resonances involving zero sign minus is solved. So, I mean, only to flash, as I said, the curvature that we are observing here is due to an effective four-level transitions. We can, I mean, formally describe this in terms of effective two-qubit Hamiltonians, two-qubit, couple-qubit Hamiltonians. I'm not going to get into the detail, but it's interesting that, depending on the kind of transition that we want to study, we have to span the phase, the Hilbert space for this effective Hamiltonian between different states, but in both cases, for both transitions, we obtain, on one case, an effective Hamiltonian for a two-coupled qubit, but coupled by an energy scale, which is the energy scale, the frequency of the resonator, okay? And on the other side, we have a coupled qubit, coupled not through the linear, I mean linear, because we are coupling through sigma set, which sigma set is the driving, I mean, operator, and in this case, they are coupling transversely. So this kind of Hamiltonians, we have been studying a lot during, I mean, the last couple of years to actually generate entanglement in between two qubits, but not in circuit-quady architectures. And what is interesting is with these two effective Hamiltonians, we reproduce mainly the exact simulations, numerical simulations from the resonance button. So, as I said, the dissipative dynamics is, I mean, is the essential key ingredient to stabilize entanglement, okay? So the protocol, as I said, we promote a transition by driving from the initial state to the one side minus state, and then by photon losses, the state decays, one photon is lost, and the state is stabilized in this fully entangled state as far as the driving is applied, this state is perfectly stable. So this is, on the left, the steady-state occupation of this entangled state. I mean, we compute the steady-state density matrix, we can compute the population, I can compute the steady-state occupation, and as I show you, this is a map of the steady-state population. You see that it reaches values close to one, okay? In this parameter region, as I mentioned before, when we stimulate our resonances to the one side minus state. And as a measure of entanglement, you use the concurrence, I'm not going to give you the formal, I mean, for those we are not, I mean, in the quantum information community, but essentially it's a measure of entanglement that gives you one when the state is fully entangled and zero with the state are separated. So there is a perfect correlation. We generate entanglement, fully entangled state, I mean, essentially this gives you essentially these two curves the same information, fully entangled state or fully concurrence equals one. So the protocol that we have proposed enables the generation of entangled bell state. Well, the temporal dynamics is in a way cumbersome, but it's interesting to see how we start with an initial state, okay? We see that at the end, the state with zero photons and maximal entangled bell state is fully populated, okay? And also it's interesting in the temporal dynamic that we see two types of oscillations. We have fast oscillations that are related to the time scale of the gap, I mean, that we have for these oscillations between up-up and down-down states, okay? That are oscillations that are of the order of 10, I mean, the period if 10 times the period of the driving, okay? So you see this fast oscillation and also the state 1 sin minus, okay? Is stimulated on time scale with the order of the inverse of the energy scale for the gaps, which is delta G, which is a much, much lower energy scale so the period is much larger, okay? You see the different time scales in these time dynamics. But I mean, at the end, we managed to populate, okay, the entangled state with population 1. So I treat a little bit in this explanation because, I mean, for the formal calculations, we neglect at first order the qubits gap because in generally, we can consider that the gaps are much, much lower than the tuning that we are considering. But in a way, we did the numerics with considering finite gaps. And these are the results. I mean, if you consider the typical gaps on sigma x on the qubits, we see that the protocol is essentially, I mean, robust against the change in the qubit gap when the gap is on sigma x. Remember that we are coupling the qubit through sigma x, okay? So in a way, this renormalized the coupling but did nothing. And in this case, if we consider a qubit with this gap in sigma y, which is, I mean, orthogonal to sigma x, we see a more, I mean, intricate pattern of generation of entanglement. But anyway, we can find white regions in the parameter space in which this protocol is robust. So to summarize, we have proposed a new protocol as far as we understand to generate and stabilize a belt state, which is essentially based on Dandau-Cennery-Stukelberg interferometry and dissipation, okay, through photon loss in the cavity in a circuit-quity architecture. So the qubits are driven with microwaves, not resonant, and beyond that, we drive in regime exploring previous protocols. And the dissipation, as I said, is mainly due to photon loss. So the requirement is to find regions in a parameter space in which transitions to the one photon psi minus state is promoted and transitions out of the zero psi minus, psi minus, I don't know, overlap. So for this, we have a trade-off between the width of the resonances and the curvature of the resonances that I show in the spectrum. So this condition give you that the delta G, which is the difference between the couples, should be much slower than G1 and G2 over omega. So I took this, I mean, from the work of World Rough many years ago in order to say, okay, this is a conservative mesh numbers that I'm going to give you, but with those numbers used in the first experiment that I show you, the condition is satisfied. And I think that it's feasible to implement this protocol. So for those of you that want more details, we have recently had this publication. So thank you. I want to thank you, all of you, my fundings and my place. These are the mountains of Bariloche. Okay, thank you. Very nice, thank you. Thank you, Maria. So the floor is open for questions. Thank you, Maha, for a beautiful talk. I have two short questions. First of all, did you try to look to the other side of a coin, namely a rabbi oscillations in the system of two qubits? Because, I mean, we are pursuing this, I mean, Stuckelberg interferometry, trying to go away from the resonant condition. Okay, but we can do it. I mean, it's a different strategy. I will explain in a coffee break why it's interesting. And the second question, did you try to look in your system of two qubits from the point of your group theory? It's a four group, which is... Yeah, yeah, yeah, yeah, yeah. It has a beautiful... This is a dynamical group, it's semi-simple. It has a hidden constant of motion, which... Okay, we didn't... I mean, I have a look at your paper, so you are going to explain to me what I'm going to do with the group theory. Because, I mean, you know, we have three and four levels, avoided crossings, but the structure of the crossings is, I mean, a little bit cumbersome, but we cannot, I mean, classify this kind of crossing, but it could be interesting to see what is the effective transitions at least analytically between these levels. Yeah. Hi, you have just that you don't go up in your spectrum when you are doing all this protocol, but with your model. But what about if you take into account the full supercondition circuit and take into account that the qubit has a higher energy state? Then I guess that you have to do this qubit that has a large anormonicity because if you do it with a trasmund, you are going to go up in the spectrum. Yeah, yeah, yeah. Actually, we work a lot with, I mean, the device from the Shostakovon flux qubit in order to investigate these Landau Center transitions, okay? So what we consider here is that we have a two qubit and we have this replica of the spectrum of the two qubits due to the photons in the cavity. But in a way, if you have more complex strategy, I think that we can excite the qubit, but what we need is to excite the qubit to a entangled state with a fixed number of photons, okay? And then if relaxation is only due to photon loss, I think that the mechanism is rose. But I mean, I don't see any difference. We are going to have different energy scales, but the trick here is to have a lowest energy scale that is given by the difference in the coupling of the qubits to the resonator, okay? We are going to have other energy scales, okay, but mostly the Landau Center transitions are going to be dominated by these small gaps. So if you have larger gaps, maybe the situation is rather complicated, but I think... Okay, we can discuss if you want later some details. Maybe I'm not catching what you are going to... I have a practical question. Is it very important that the drive of the two qubits are perfectly in phase? No. No, actually, we have explored with different detunions. Actually, I mean, instead of being in phase, we move one of the qubits from, I mean, to detuning positive and the other little bit. And there is a range of detuning in which the protocol still works, okay? The spectrum is much rather involved because it's not so symmetric. You have different slopes, I mean, if you have different qubits, but this is the first and easy, but I think that we can explore those. All right, if there are no more questions, then let's thank Maria again. We started a bit late and we ran a bit over time, so the coffee break would like to...