Quantum dynamics and quantum control of spins in diamond
Dobrovitski, Viatechslav
One of the most interesting challenges in modern physics is to understand, manipulate, and control materials and processes at the level of a single quantum spin. Beside fundamental interest, this research is important for applications ranging from spintronics and information processing to high-precision metrology. Nitrogen-vacancy (NV) impurity centers in diamond have recently emerged as a unique platform for investigating quantum dynamics, decoherence, and quantum control of single spins in solid-state environments. NV centers demonstrate an unusual combination of spin-dependent optical properties, individual addressability, and long spin coherence times. The NV spin state can be manipulated both optically and magnetically, and the quantum control operations can be performed with very high fidelity (>99%). Due to these uniquely favorable properties, quantum dynamics of a single NV spin can be investigated in great detail. I will discuss quantum dynamics of the NV centers, and the decohering effect of the spin bath (made of atomic nitrogen impurities) on the evolution of a NV spin. I will demonstrate how the decoherence dynamics depends on the experimental conditions. Further, I will discuss how the modern techniques of quantum control and dynamical decoupling can be employed to preserve coherence of quantum spins. Using a variety of analytical and numerical tools, we can characterize and optimize the factors which limit controllability. Finally, I will demonstrate how the theoretically optimized protocols for quantum control have been used in order to extend coherence time of individual NV spins, and how dynamical decoupling can be integrated with the gate operation in the prototype hybrid system, the electron-nuclear spin register.
 All right, good afternoon. We're ready to start the next session. But before we do just a quick announcement about the banquet, the banquet will be here downstairs where we had lunch on the first day before they moved us outside. And everyone's welcome. So our first speaker is Slavoj Dobrowicki. And he will tell us about quantum dynamics and quantum control of spins and diamonds. And so, thank you very much for coming, for finishing lunch early, coming to my talk. So I will be talking about a few things we have done with our theoretical collaborators, with my former post-doctor Gichlwam, who is now working with Daniel, with the group of Ronald Hansen from Technical University of Delft and from the group of David Arshalom, two hours away from... So recently there have been a lot of interest in individual quantum spins in solid state. And people have studied a lot and some of the systems are probably known to you. These are quantum dots, which have been popular, very popular, for the last two years. Donors and silicon, where they finally achieved the ability to work with a single donor center. And the subject of my talk, and the center in diamond. So why do we care about individual spins? And so let's stay to all four of you, for reasons. Because they're good for at least small to medium scale quantum information processing. They are good for coherent spin electronics and photonics, and this is probably the most fast-growing area of their potential applications. High precision metrology and magnetic sensing at nano-scale. And huge area with very, very impressive potential applications. But of course, as always, we are facing a grand challenge, how to control single quantum spins in solids. And for this, we need to solve quite a few fundamental problems, to understand dynamics of individual quantum spin, to understand how to control it, how to preserve its coherence, and how to generate and preserve entanglement between quantum spins. And for all these, so this is the picture which in the shows what we want to do. We want to be able to pick up a single spin in solid state, do whatever we want, and whatever we want. And spins and diamond, they constitute an excellent test bed for this. They have long coherence time. They are individually addressable and controllable both optically and magnetically. And at our disposal, we have an excellent menu of dynamical decoupling protocols. And just to start with the traditional introduction, usually they are analyzed and classified using Magnus expansion, when we have terms of zero order, first order, second order, and so on, assuming that the distance between the dynamical decoupling pulses is very small. And the simplest one is periodic dynamical decoupling, where when we wait, apply pulse, wait, apply pulse, tau x, tau x. We can symmetrize it, so there has been a lot of introduction. So we can have this CPMG protocol, wait, apply pulse, wait twice, apply pulse, wait. And we can concatenate them, put them inside each other, like this Russian doll, and achieve more impressive protocols. However, for any, probably for any theorist who starts working with experimentalists, there is a moment of truth when you realize that whatever all these fancy and beautiful theories you're doing, they're totally useless, and you have to deal with something much more mundane, something like pulse errors, or the fact that this generator doesn't have enough dynamical range, and all your beautiful suggestions, they either go down the drain, or you have to do something else. So what are the problems with the standard approach? First, we always hear about the norm of certain Hamiltonians. Well, reality, all these norms grow with the size of the bath. So whenever we write this beautiful expression, norm of H multiplied by tau, don't forget that this is essentially infinite. And it is not small parameter, it's actually infinite parameter. Second, validity conditions in reality are not satisfied, because the ultraviolet cutoff is usually too large, but still, well, no DD works. Behavior at long times is not quite clear, because again, of this nature of the cumulant expansion, and the cumulation of pulse errors and imperfections is also not well understood. So this is why we have to usually do some numerical stuff to really help experimentally and assess how the codes will perform in reality. And so we can do it this way or that way, and often we can do much simpler things. They may be not as impressive from point of view, but they produce really good results, such as mintial techniques. So what will be my talk about? First, about quantum control and dynamical decoupling of an e-center. There lately has been spectacular progress. There were three papers published in different journals, probably within the time span of one month or so. At least I remember how Ronald called me at some point and said that Corey put the paper on archive and we have to put our paper on archive, so it was, but it was fun. Second, how to do the coherence protected quantum gates, and I will demonstrate you the coherence protected quantum algorithm in solid state, and correct me if I'm wrong, but as far as I know, this is the first two-qubit quantum computation with individual solid state speed, which we finally achieved. So what are the characters in this story? The simplest impurity in diamond is a substitutional nitrogen. We take away carbon, we put nitrogen instead, we have one excess spin, so there is a spin one half. And these guys, green guys are bad. They constitute our spin bath, our environment, which decahiers us, and they're coupled to each other with long-range dipolar coupling. But when nitrogen meets vacancy, the transformation happens, the bad guy becomes a good guy, we have this beautiful nitrogen vacancy center, which is our central spin system, or our qubit. The red one is the electron spin, which were after, actually it's a spin of two holes, so the total spin is one, and there is also a nuclear spin of nitrogen 14, it will be nuclear spin one or nitrogen 15, spin one half. And they're coupled on-site, and it will play an important role later. And this center is coupled by the polar interaction to a large number of nitrogens. So what's so good about this center? It can be initialized, manipulated, and readout, optically and magnetically. So we can look at a single individual spin in a solid state. Why is it possible there is a very interesting and not even completely understood level diagram, but to make a long story short, we have spin one, we have three sub-levels, 0 plus 1 minus 1. Zero sub-level, if we shine a photon on it, it gets excited to the excited state, stays there for 10 nanoseconds, and falls back and emits a photon. So every time, photon in, photon out, photon in, photon out. But if we're in one of those states, we can be excited, then make what is called in this quantum chemical jargon intersystem crossing, and we get stuck in this shelving state for a long, long time, for 300 nanoseconds, almost infinity. So, and even when we come back, we come back first to m equals zero state. So we started here, but eventually we came back here. And when there is a transition from here to here, no visible photons are emitted. So if we're in m equals zero, photon in, photon out, photon in, photon out. If we're in these two states, photon in, nothing. So, and even when there's nothing over, we're polarized in m equals zero state with typical polarization of order of 90% or higher, which is... So, initialization and redoubt simply by shining a green laser. Nothing else is needed. Now, let's look at the coherence. So here we have this nitrogen vacancy center. We apply some magnetic field. We forget about this level, minus one. It's not of interest for us, it's an idle. And here is our qubit. And we have buff spins, nitrogen atoms. But because of the different parameters, this resonant frequency is very different from the frequency of this transition. So what we're dealing with is a pure defacing. There is just effective level splitting between these two qubit levels. And there is some operator which describes time-dependent action of the bath on the qubit. Purely defacing term. Now, how to deal with this? Of course, if we want to make our... If we want to be completely honest, it's a many-body quantum non-equilibrium problem which we have no way to address analytically, at least unless we have some good parameters, which we don't in this case. But in this particular case, it appears that due to the fact that our bath feels negligible back action from the central spin, we can replace the field created by the bath just as a random C-number field, which is Gaussian stationary and Markovian noise, or the null and back process, which is characterized by two parameters only. Strength of the noise, B, which is determined by the spin-bath coupling between the V and the bath of nitrogens. And the correlation time, which is determined by the coupling between nitrogens, between rings. And we can do direct many-spin modeling. We can run simulations with 20 spins. We can run simulations with 10,000 spins. And again, to make a long story short, we just can confirm that indeed this very simple model is good enough, is adequate for our experiments. Here is the correlation function. Here is what we get for spin echo, just taking numerical simulation and versus analytical model free evolution. So you see agreement is very good. So, qubit in a Gaussian stationary Markovian mean field. Now, just to complete the story, we can look at the experimental Frida coherence. These red dots are experiment, gray line theory. We can extract C2 star time, and therefore we can extract the coupling of the qubit to the environment. Here you see modulation induction signal. It is because of hyperfine coupling of the nuclear spin. Next, just to make sure that our mean field model is indeed correct, we can do spin echo. And we see that it indeed decays as theory predict, e2 minus t cube decay. And the correlation time of the both is 25 microseconds, which is long enough for us to do dynamical decoupling. So, our first step, quantum control and dynamical decoupling. Extending coherence time of a single MV center. First, of course, since nobody has done it before, on MV centers before us, the question was always which protocol to choose from this plethora of protocols. And we started analyzing one after another. First, we started with PDD, like simplest protocol, the most obvious choice. And we ran into the problem right away. In short times, it decays four times faster than we would like to. Only at longer times, it becomes here instead of four thirds, we have one third. So, it's good, it's simple, but unfortunately it decays fast initially. Not what we want. So, let's try CPMG. It appears that CPMG has this slow decay at all times. Beautiful. Now, of course, we want to try to concatenate it. And we see that fast decay at all times, if we concatenate PDDs, and if we concatenate CPMGs, slow decay at all times, and no improvement. Absolutely no improvement, which means that we will be just programming a more difficult sequence without any actual, any advantage. So, why? Well, first of all, it's kind of an obvious statement, but I want to reiterate it, that coherence time can be extended well beyond 25 microseconds, well beyond the correlation time of the bus. Second, Magnus expansion gives us completely wrong predictions, because instead of this decay for PDD or for CPMG, we have, instead of tau to the fourth, we have tau to the cube. Symmetrization or concatenation gives no improvement, and simply because we have Orson-Nullenbeck noise, with a heavy tail, one over omega squared. Second moment of this noise is formally infinite, and it corresponds to what I started with, that the norm of the butt Hamiltonian is infinite. Actually, it's not quite infinite, but it's at 5 gigahertz, and it's much faster than anything, any other timescale in our experiments. So, we know the protocol, CPMG. We start running it, and we immediately see that X component is protected very well, and Y component is not protected very well. So, what's wrong? Because also, obvious control pulses are not perfect. And it's actually interesting to see why they are not perfect. When we're trying to control our envy center with the actual experimental situation, you see that during the whole pulse, we have only one, two, three oscillations, and we have a ring down. So, again, all these beautiful rotation frame approximation, inapplicable. If we have very slow driving, then we indeed see what we expect, to see beautiful rib oscillations of the spin. But when the driving becomes stronger, because this is what we need to achieve very short pulses, you see that we don't have the, you know, the same kind of oscillations, but we have a very short pulse, and we have a very short pulse, and we achieve very short pulses. You see that we don't have these beautiful nice oscillations. Instead, we have kind of spin stalls in some states, then quickly jumps from up to down, stays there for a while, jumps halfway, then stays, and shows the sequence of past jumps and stalls. So, and this is not an experimental noise, as you see it's reproduced completely by simulation. This is just what happens when we try to drive our spin very fast. And of course, it introduces a lot of errors in the pulses, and we need to characterize them, so we did some work how to develop the scheme for character of the pulse errors. Then we should study how the errors accumulate. We also did that, and we saw some interesting stuff. I hope Kavi will talk more about this. So, again, bottom line is that if we apply pulses along X, X component is preserved well, why not? But if we apply pulses along X and Y, and use these X, Y, 4 sequence, which was introduced yesterday, we see that X and Y are protected well, and coherence time is indeed extended. So, here you see that the coherence time is in the region of about 7 milliseconds, although we started with 2. Good. We can look at other a pyrotic sequences, UDD, QDD. So, what do we see? Well, first we do not expect them to work very well, because we have no hard cutoff in the buff, and indeed they do not perform as well as we expected. But there is a deeper problem with them if we look at the robustness to the pulse errors. And here are the simulations with imperfect pulses, whether the parameters of every pulse are taken from experiment. So, as you see, still X, Y, 4 performs much better than QDD, and of course incomparably better than UDD. So, robustness to pulses plays an important role. But, again, if we choose our protocol correctly, if we take into account the errors, and if we do things clear, in a clean manner, then we see exactly what theory predicts. And what does it predict? First, for any number of pulses, the decay should have this form, E to minus T cubed. So, if we take spin echo, 4 pulses, 8 pulses, 136 pulses, and rescale them, they should fall on the same master curve, and this is exactly what you see. And if we measure the coherence time as a function of the number of pulses, it should demonstrate N to the 2 thirds. And, again, this is exactly what experiment shows. So, applying 136 pulses, coherence is increased by a factor of almost 30, we achieve 90 microsecond, which is great. And, well, actually, I wrote here no limit inside, we're limited at this time scale, we started being limited by T1 processes. So, we're practically pushed dynamical decoupling to the limit. But this is what happened last year. Since then, we're doing some other things, using dynamical decoupling to improve single-spin magnetometry, or to do the detailed probing of the mesoscopic spin bath, which surrounds a single NV center. So, we take a single center and study around 10 spins, which form an effective bath for these particular centers. So, essentially, again, it gives us ability to study mesoscopic spin dynamics. But this is only half of the story. Protecting spins is good, but we want to do something useful with them. And for this, we need many NVs to talk to each other, to process information in some way. And there were one particularly useful ways to make NVs talk to each other, is to use hybrid systems. Inject systems are very popular now, because they combine best features of different qubits, which we know about. So, here in particular, there are suggestions to put NVs on conciliators and use the nanomechanical oscillators as data bus between the NV centers. There's been a suggestion to use spin chains, but the simplest one, which you can think of, and the most probably widespread, is when we use electron spin as a processor for quantum information, and nuclear spin as a memory. So, anyway, we have many spins, and they decay here. We want to do something useful with them. So, if we think about standard mode of quantum operation, of course, it doesn't work. What we can do is unprotected quantum gate, but then again protected. Or we can do protected storage, but then we kill the coupling, not only between the different qubits, which have different time scales, but also between our qubit and the bus. So, we can efficiently preserve the state. We can efficiently isolate our NV center. But in order to perform some interesting computation, we need to change the state. We need to keep electron spin, for instance, talking to the nuclear spin. This is what we want to achieve. And to do this, we need to be able to combine dynamical decoupling pulses with the gate operation in the seamless fashion. So, decouple and compute at the same time. And again, since we are dealing with hybrid system, different qubits have very different coherence and control time scales. So, our electron, we start decoupling our electron, nuclear spin just barely starts to move. So, by the time the electron spins, it occurs, nuclear spin just starts to rotate. If we... So, if we try to control the nuclear spin, it takes much longer than the electron decoherence time. So, we have essentially the choice. Either decouple the electron, apply dynamical decoupling, but then no gates possible because we decouple it from the nucleus. Or we can do gating without dynamical decoupling. But then again, we don't have any gates. So, there is... Of course, I wouldn't talk about this if there was no way out. And the way out is to use the resonance in the system. This... What... I don't know, for this talk, I called it guard. Gates with resonance decoupling. So, cut our system in more detail. In the rotating frame, this is the Hamiltonian which describes it. SZ is a very fast, decayed electron spin. I is the very, very slow, barely moving nuclear spin. So, you can see, for instance, that here we have 2 MHz hyperfine coupling, but the driving we apply to the nuclear spin is only 18 kHz. And this is a very typical figure. So, let's look what happens in the rotating frame. If the electron spin is on a state 1, SZ equals 1, and we have nuclear rotating around Z, because this term kills that one. If the electron is on a state 0, this term is absent, we have only driving. The nucleus slowly, slowly rotates around the X axis. So, the only problem here is that the electron switches very fast between Z. This is the essence of dynamical decoupling. And... But the nuclear spin should keep track of what was the electron state long before we started decoupling. At first sight, it's kind of contradictory, but in fact, if we look at the motion of the nuclear spin, if we start from the state 0, it will be 0, pulse on the electron spin, so the electron spin will be 0, 1, 1, 0. If we start at the electron spin on state 1, it will be state 1, 0, 0, 1. And in general, the Hamiltonians are different, but these rotation axes for the nuclear spin, they are closed. They are very close to each other, simply because they are both close to the Z axis. This is just a formal expression. So, you see that the product is almost 1 minus some small parameter. And fortunately, resonance, when this something becomes small, and to understand what happens at this case, we can just look at how spin rotates. In this case, we have, during the time tau, we have exactly half a rotation around the Z axis. So, if we started with the electron in state 1, it rotates pi around Z, then x around the x axis, and then another pi around Z. So, as a result, we have minus x of the nuclear spin. If the electron started in a state 0, then we have rotation around x, full rotation around Z, which does nothing, and then again, rotation around x. So, our out state is plus x. That's it. That's a resonance. And now we can do something good. But first, we should understand that the resonance is narrow. Here is just the graph of these functions I showed before. Timing jitter should be less than 1 nanosecond over 100 microsecond time span. So, this is very precise requirement should be satisfied to do this. If we make an error by 10 nanosecond, the fidelity drops to 90% already. But there is also a beautiful thing that if we're in this resonance, we do nuclear spin conditional rotation. But if we're exactly integer number of full rotations away from the resonance, then we have unconditional nuclear spin rotation, which is not interrupted by interaction with the electron. So, we can do all nuclear gates only by changing the inter-pulse time delay. And first proof of concept. It first is protected controlled rotation gate. And here you see if the electron starts in the spin 1, the nucleus oscillates around the x-axis. But if we are in the state 0, then nucleus stays in place and shows only a little bit of T1 decay. And this is the state tomography, which shows us that the fidelity of the state, of the bell state created in this way, is 97%. Although we are about 10 times longer than the electron, the de-coherence time for induction decay time. Now, we can do something more interesting. T2 star is just a defasing. We can show that we can go beyond the T2 time and do still high fidelity gates. So, what's been done here, T2, 50 microsecond, gate time is longer. What we want to do is to create the bell state. And again, we can do it with four decoupling pulses on the electron, eight decoupling, 16 decoupling pulses on the electron. But the only thing we need to care about is keeping precisely the resonant condition. And when we do it, you see for 16 pulses, we're doing in the same way as if no de-coherence was present here. So good. So we have a working gate. We decoupled the electron. We created the de-coherence protected gate for the nuclear spin. We're doing it on a time scale which is twice larger than T2. So let's do some algorithm. Let's... oops. Let's see. Yes, it works. Good. So we can take two qubits. We have electron spin and the nuclear spin. Grover's algorithm converges in one iteration, but still, if we look at the total time needed to perform the algorithm, it is 300 microsecond. And our T2 time is only 250 microsecond. So this is the block diagram. And this is exactly the scheme of the pulses applied in the electron channel and in the nuclear channel. I just told you we just keep driving very slowly with very small driving. And we just keep the... adjust the distance between the electron decoupling pulses, either in-perme or half-integer number of rotations. And yes, if we look at the Grover's algorithm, for instance, if we want our target state is one up, we obtain it with the fidelity 95%. And here are just the experimental snapshots of different stages of the Grover algorithm. First, we prepare the equals proposition, flip, do the inversion, flip again, and restore the state. So the fidelity for the target state one up is 95%. For the other states, it's 93, 92, 91. But in all cases, we're above 90%. Although, again, our algorithm time is 60% longer than the spin echo. So we can achieve high fidelity beyond coherence, which is great. So just as a conclusion, the diamond-based quantum information processing becomes competitive, because not only our work, but in general the community working in this area, demonstrated already spin photon entanglement, protected gates, and dynamical decoupling. So we have all three crucial ingredients, at least for small-scale quantum information processing. Coherence time can be extended. We show 25-fold enhancement. Other people show comparable figures. Dynamical decoupling can be efficiently combined with gates. And we can even do protected algorithms at the times which exceed T2 times, spin echo time, with the fidelity above 90%. And just as a side note, I think it's nice that this was the first two-qubit computation on individually-addressed state spins. So thank you for your attention. Thank you, Slava, for finishing five minutes earlier. Got us back on schedule, but we do have some time for questions. So maybe it sounds stupid, but what if you wanted to go beyond T1? We need to decouple phonons. In principle, it can be done. But there are other ways to beat that limit. So just right now it's not the actual limit, because by lowering temperature even to 20K, you have T1 in minutes, a water of minutes. I'm sorry, I forgot to say the most important thing. This is all done at room temperature. I was giving this talk to the mostly diamond audiences. I forgot to mention the crucial thing. This is all room temperature. So essentially what you have, you have just an objective, one laser, and a piece of diamond, which you can buy for $40. And that's it. You have a quantum computer. No, I'm serious. I'm not joking. Well, yes, you need microfabrication lab somewhere. How many carats diamond is this? Oh, it's about this big, about couple millimeters by couple millimeters. But don't show it to anybody. It's very ugly. It looks like a piece of old plastic. It has a lot of nitrogen in it. It's yellow. This is very nice. So how about you go to like a more than two qubits? What would be the perspective? More than two qubits? Well, again, first weekend there is a very successful effort, which recently also there was Nature Physics published last year, where people were able to place two and well, place maybe. Okay, there were two NV centers which were close enough to appreciably interact via dipolar interactions. But at the same time, far enough to be resolved optically. So there has been already demonstration of two dipolar coupled, individually addressable, and these centers. Actually, if I had my computer, I even have a little slide on this, because we did the simulations for this. We also showed that we can go far, far beyond T2 time also. So the carats protected gates can be done in this case. For that case, is it also room temperature? That's limited at T1, T2, T3? No, no, no, no. I think it was something like 50 kilo. I'm not sure. Let me. But it can be room temperature as well. For this system, the difference is in T1, nothing else. And maybe, let me add one more. There has been spin photon entanglement demonstrated last year. So just entangling, probabilistic entanglement between two NVs is just a matter of time. Whoever does it first actually does. So why do you use this ugly diamond with lots of nitrogen and not a clean one with very few nitrogens? Well, we use the clean one too. The one with 250 microsecond coherence time, it was a clean one. But it doesn't really matter. We can do well cases. Yeah, but if you want to get long coherence time, it's easier. Yes, we did this. The one with where the algorithm was done, 250 microsecond. It was a clean one. Well, you still need to decouple carbon-13 spins. So you cannot go without decoupling anyway. Thanks again.
 All right, good afternoon.  We're ready to start the next session.  But before we do just a quick announcement about the banquet,  the banquet will be here downstairs where we had lunch on the first day  before they moved us outside.  And everyone's welcome.  So our first speaker is Slavoj Dobrowicki.  And he will tell us about quantum dynamics and quantum control of spins and diamonds.  And so, thank you very much for coming, for finishing lunch early, coming to my talk.  So I will be talking about a few things we have done with our theoretical collaborators,  with my former post-doctor Gichlwam, who is now working with Daniel,  with the group of Ronald Hansen from Technical University of Delft  and from the group of David Arshalom, two hours away from...  So recently there have been a lot of interest in individual quantum spins in solid state.  And people have studied a lot and some of the systems are probably known to you.  These are quantum dots, which have been popular, very popular, for the last two years.  Donors and silicon, where they finally achieved the ability to work with a single donor center.  And the subject of my talk, and the center in diamond.  So why do we care about individual spins?  And so let's stay to all four of you, for reasons.  Because they're good for at least small to medium scale quantum information processing.  They are good for coherent spin electronics and photonics,  and this is probably the most fast-growing area of their potential applications.  High precision metrology and magnetic sensing at nano-scale.  And huge area with very, very impressive potential applications.  But of course, as always, we are facing a grand challenge,  how to control single quantum spins in solids.  And for this, we need to solve quite a few fundamental problems,  to understand dynamics of individual quantum spin,  to understand how to control it, how to preserve its coherence,  and how to generate and preserve entanglement between quantum spins.  And for all these, so this is the picture which in the shows what we want to do.  We want to be able to pick up a single spin in solid state, do whatever we want, and whatever we want.  And spins and diamond, they constitute an excellent test bed for this.  They have long coherence time.  They are individually addressable and controllable both optically and magnetically.  And at our disposal, we have an excellent menu of dynamical decoupling protocols.  And just to start with the traditional introduction,  usually they are analyzed and classified using Magnus expansion,  when we have terms of zero order, first order, second order, and so on,  assuming that the distance between the dynamical decoupling pulses is very small.  And the simplest one is periodic dynamical decoupling,  where when we wait, apply pulse, wait, apply pulse, tau x, tau x.  We can symmetrize it, so there has been a lot of introduction.  So we can have this CPMG protocol, wait, apply pulse, wait twice, apply pulse, wait.  And we can concatenate them, put them inside each other, like this Russian doll,  and achieve more impressive protocols.  However, for any, probably for any theorist who starts working with experimentalists,  there is a moment of truth when you realize that whatever all these fancy and beautiful  theories you're doing, they're totally useless,  and you have to deal with something much more mundane, something like pulse errors,  or the fact that this generator doesn't have enough dynamical range,  and all your beautiful suggestions, they either go down the drain,  or you have to do something else.  So what are the problems with the standard approach?  First, we always hear about the norm of certain Hamiltonians.  Well, reality, all these norms grow with the size of the bath.  So whenever we write this beautiful expression, norm of H multiplied by tau,  don't forget that this is essentially infinite.  And it is not small parameter, it's actually infinite parameter.  Second, validity conditions in reality are not satisfied,  because the ultraviolet cutoff is usually too large, but still, well, no DD works.  Behavior at long times is not quite clear, because again, of this nature of the cumulant expansion,  and the cumulation of pulse errors and imperfections is also not well understood.  So this is why we have to usually do some numerical stuff to really help experimentally  and assess how the codes will perform in reality.  And so we can do it this way or that way, and often we can do much simpler things.  They may be not as impressive from point of view, but they produce really good results, such as mintial techniques.  So what will be my talk about?  First, about quantum control and dynamical decoupling of an e-center.  There lately has been spectacular progress.  There were three papers published in different journals, probably within the time span of one month or so.  At least I remember how Ronald called me at some point and said that Corey put the paper on archive  and we have to put our paper on archive, so it was, but it was fun.  Second, how to do the coherence protected quantum gates,  and I will demonstrate you the coherence protected quantum algorithm in solid state,  and correct me if I'm wrong, but as far as I know, this is the first two-qubit quantum computation  with individual solid state speed, which we finally achieved.  So what are the characters in this story?  The simplest impurity in diamond is a substitutional nitrogen.  We take away carbon, we put nitrogen instead, we have one excess spin, so there is a spin one half.  And these guys, green guys are bad.  They constitute our spin bath, our environment, which decahiers us,  and they're coupled to each other with long-range dipolar coupling.  But when nitrogen meets vacancy, the transformation happens,  the bad guy becomes a good guy, we have this beautiful nitrogen vacancy center,  which is our central spin system, or our qubit.  The red one is the electron spin, which were after, actually it's a spin of two holes,  so the total spin is one, and there is also a nuclear spin of nitrogen 14,  it will be nuclear spin one or nitrogen 15, spin one half.  And they're coupled on-site, and it will play an important role later.  And this center is coupled by the polar interaction to a large number of nitrogens.  So what's so good about this center?  It can be initialized, manipulated, and readout,  optically and magnetically.  So we can look at a single individual spin in a solid state.  Why is it possible there is a very interesting and not even completely understood level diagram,  but to make a long story short, we have spin one, we have three sub-levels, 0 plus 1 minus 1.  Zero sub-level, if we shine a photon on it, it gets excited to the excited state,  stays there for 10 nanoseconds, and falls back and emits a photon.  So every time, photon in, photon out, photon in, photon out.  But if we're in one of those states, we can be excited,  then make what is called in this quantum chemical jargon intersystem crossing,  and we get stuck in this shelving state for a long, long time, for 300 nanoseconds, almost infinity.  So, and even when we come back, we come back first to m equals zero state.  So we started here, but eventually we came back here.  And when there is a transition from here to here, no visible photons are emitted.  So if we're in m equals zero, photon in, photon out, photon in, photon out.  If we're in these two states, photon in, nothing.  So, and even when there's nothing over, we're polarized in m equals zero state  with typical polarization of order of 90% or higher, which is...  So, initialization and redoubt simply by shining a green laser.  Nothing else is needed.  Now, let's look at the coherence.  So here we have this nitrogen vacancy center.  We apply some magnetic field.  We forget about this level, minus one.  It's not of interest for us, it's an idle.  And here is our qubit.  And we have buff spins, nitrogen atoms.  But because of the different parameters, this resonant frequency is very different  from the frequency of this transition.  So what we're dealing with is a pure defacing.  There is just effective level splitting between these two qubit levels.  And there is some operator which describes time-dependent action of the bath on the qubit.  Purely defacing term.  Now, how to deal with this?  Of course, if we want to make our...  If we want to be completely honest, it's a many-body quantum non-equilibrium problem  which we have no way to address analytically, at least unless we have some good parameters,  which we don't in this case.  But in this particular case, it appears that due to the fact that our bath feels negligible  back action from the central spin, we can replace the field created by the bath  just as a random C-number field, which is Gaussian stationary and Markovian noise,  or the null and back process, which is characterized by two parameters only.  Strength of the noise, B, which is determined by the spin-bath coupling between the V and the bath of nitrogens.  And the correlation time, which is determined by the coupling between nitrogens, between rings.  And we can do direct many-spin modeling.  We can run simulations with 20 spins.  We can run simulations with 10,000 spins.  And again, to make a long story short, we just can confirm that indeed this very simple model is  good enough, is adequate for our experiments.  Here is the correlation function.  Here is what we get for spin echo, just taking numerical simulation and versus analytical model free evolution.  So you see agreement is very good.  So, qubit in a Gaussian stationary Markovian mean field.  Now, just to complete the story, we can look at the experimental Frida coherence.  These red dots are experiment, gray line theory.  We can extract C2 star time, and therefore we can extract the coupling of the qubit to the environment.  Here you see modulation induction signal.  It is because of hyperfine coupling of the nuclear spin.  Next, just to make sure that our mean field model is indeed correct, we can do spin echo.  And we see that it indeed decays as theory predict, e2 minus t cube decay.  And the correlation time of the both is 25 microseconds, which is long enough for us to do dynamical decoupling.  So, our first step, quantum control and dynamical decoupling.  Extending coherence time of a single MV center.  First, of course, since nobody has done it before, on MV centers before us,  the question was always which protocol to choose from this plethora of protocols.  And we started analyzing one after another.  First, we started with PDD, like simplest protocol, the most obvious choice.  And we ran into the problem right away.  In short times, it decays four times faster than we would like to.  Only at longer times, it becomes here instead of four thirds, we have one third.  So, it's good, it's simple, but unfortunately it decays fast initially.  Not what we want.  So, let's try CPMG.  It appears that CPMG has this slow decay at all times.  Beautiful.  Now, of course, we want to try to concatenate it.  And we see that fast decay at all times, if we concatenate PDDs,  and if we concatenate CPMGs, slow decay at all times, and no improvement.  Absolutely no improvement, which means that we will be just programming  a more difficult sequence without any actual, any advantage.  So, why?  Well, first of all, it's kind of an obvious statement, but I want to reiterate it,  that coherence time can be extended well beyond 25 microseconds,  well beyond the correlation time of the bus.  Second, Magnus expansion gives us completely wrong predictions,  because instead of this decay for PDD or for CPMG, we have,  instead of tau to the fourth, we have tau to the cube.  Symmetrization or concatenation gives no improvement,  and simply because we have Orson-Nullenbeck noise,  with a heavy tail, one over omega squared.  Second moment of this noise is formally infinite,  and it corresponds to what I started with,  that the norm of the butt Hamiltonian is infinite.  Actually, it's not quite infinite, but it's at 5 gigahertz,  and it's much faster than anything, any other timescale in our experiments.  So, we know the protocol, CPMG.  We start running it, and we immediately see that X component is protected very well,  and Y component is not protected very well.  So, what's wrong?  Because also, obvious control pulses are not perfect.  And it's actually interesting to see why they are not perfect.  When we're trying to control our envy center with the actual experimental situation,  you see that during the whole pulse, we have only one, two, three oscillations,  and we have a ring down.  So, again, all these beautiful rotation frame approximation,  inapplicable.  If we have very slow driving, then we indeed see what we expect,  to see beautiful rib oscillations of the spin.  But when the driving becomes stronger, because this is what we need to achieve very short pulses,  you see that we don't have the, you know,  the same kind of oscillations,  but we have a very short pulse,  and we have a very short pulse,  and we achieve very short pulses.  You see that we don't have these beautiful nice oscillations.  Instead, we have kind of spin stalls in some states,  then quickly jumps from up to down, stays there for a while,  jumps halfway, then stays,  and shows the sequence of past jumps and stalls.  So, and this is not an experimental noise,  as you see it's reproduced completely by simulation.  This is just what happens when we try to drive our spin very fast.  And of course, it introduces a lot of errors in the pulses,  and we need to characterize them,  so we did some work how to develop the scheme for character of the pulse errors.  Then we should study how the errors accumulate.  We also did that,  and we saw some interesting stuff.  I hope Kavi will talk more about this.  So, again, bottom line is that  if we apply pulses along X,  X component is preserved well, why not?  But if we apply pulses along X and Y,  and use these X, Y, 4 sequence, which was introduced yesterday,  we see that X and Y are protected well,  and coherence time is indeed extended.  So, here you see that the coherence time is in the region of about 7 milliseconds,  although we started with 2.  Good.  We can look at other a pyrotic sequences, UDD, QDD.  So, what do we see?  Well, first we do not expect them to work very well,  because we have no hard cutoff in the buff,  and indeed they do not perform as well as we expected.  But there is a deeper problem with them  if we look at the robustness to the pulse errors.  And here are the simulations with imperfect pulses,  whether the parameters of every pulse are taken from experiment.  So, as you see, still X, Y, 4 performs much better than QDD,  and of course incomparably better than UDD.  So, robustness to pulses plays an important role.  But, again, if we choose our protocol correctly,  if we take into account the errors,  and if we do things clear, in a clean manner,  then we see exactly what theory predicts.  And what does it predict?  First, for any number of pulses, the decay should have this form,  E to minus T cubed.  So, if we take spin echo, 4 pulses, 8 pulses, 136 pulses,  and rescale them, they should fall on the same master curve,  and this is exactly what you see.  And if we measure the coherence time as a function of the number of pulses,  it should demonstrate N to the 2 thirds.  And, again, this is exactly what experiment shows.  So, applying 136 pulses, coherence is increased by a factor of almost 30,  we achieve 90 microsecond, which is great.  And, well, actually, I wrote here no limit inside,  we're limited at this time scale, we started being limited by T1 processes.  So, we're practically pushed dynamical decoupling to the limit.  But this is what happened last year.  Since then, we're doing some other things,  using dynamical decoupling to improve single-spin magnetometry,  or to do the detailed probing of the mesoscopic spin bath,  which surrounds a single NV center.  So, we take a single center and study around 10 spins,  which form an effective bath for these particular centers.  So, essentially, again, it gives us ability to study mesoscopic spin dynamics.  But this is only half of the story.  Protecting spins is good, but we want to do something useful with them.  And for this, we need many NVs to talk to each other,  to process information in some way.  And there were one particularly useful ways to make NVs talk to each other,  is to use hybrid systems.  Inject systems are very popular now,  because they combine best features of different qubits, which we know about.  So, here in particular, there are suggestions to put NVs on conciliators  and use the nanomechanical oscillators as data bus between the NV centers.  There's been a suggestion to use spin chains,  but the simplest one, which you can think of, and the most probably widespread,  is when we use electron spin as a processor for quantum information,  and nuclear spin as a memory.  So, anyway, we have many spins, and they decay here.  We want to do something useful with them.  So, if we think about standard mode of quantum operation, of course, it doesn't work.  What we can do is unprotected quantum gate, but then again protected.  Or we can do protected storage, but then we kill the coupling,  not only between the different qubits, which have different time scales,  but also between our qubit and the bus.  So, we can efficiently preserve the state.  We can efficiently isolate our NV center.  But in order to perform some interesting computation, we need to change the state.  We need to keep electron spin, for instance, talking to the nuclear spin.  This is what we want to achieve.  And to do this, we need to be able to combine dynamical decoupling pulses  with the gate operation in the seamless fashion.  So, decouple and compute at the same time.  And again, since we are dealing with hybrid system,  different qubits have very different coherence and control time scales.  So, our electron, we start decoupling our electron,  nuclear spin just barely starts to move.  So, by the time the electron spins, it occurs, nuclear spin just starts to rotate.  If we...  So, if we try to control the nuclear spin, it takes much longer than the electron decoherence time.  So, we have essentially the choice.  Either decouple the electron, apply dynamical decoupling,  but then no gates possible because we decouple it from the nucleus.  Or we can do gating without dynamical decoupling.  But then again, we don't have any gates.  So, there is...  Of course, I wouldn't talk about this if there was no way out.  And the way out is to use the resonance in the system.  This... What... I don't know, for this talk, I called it guard.  Gates with resonance decoupling.  So, cut our system in more detail.  In the rotating frame, this is the Hamiltonian which describes it.  SZ is a very fast, decayed electron spin.  I is the very, very slow, barely moving nuclear spin.  So, you can see, for instance, that here we have 2 MHz hyperfine coupling,  but the driving we apply to the nuclear spin is only 18 kHz.  And this is a very typical figure.  So, let's look what happens in the rotating frame.  If the electron spin is on a state 1, SZ equals 1,  and we have nuclear rotating around Z, because this term kills that one.  If the electron is on a state 0, this term is absent, we have only driving.  The nucleus slowly, slowly rotates around the X axis.  So, the only problem here is that the electron switches very fast between Z.  This is the essence of dynamical decoupling.  And... But the nuclear spin should keep track of what was the electron state long before we started decoupling.  At first sight, it's kind of contradictory, but in fact, if we look at the motion of the nuclear spin,  if we start from the state 0, it will be 0, pulse on the electron spin,  so the electron spin will be 0, 1, 1, 0.  If we start at the electron spin on state 1, it will be state 1, 0, 0, 1.  And in general, the Hamiltonians are different, but these rotation axes for the nuclear spin, they are closed.  They are very close to each other, simply because they are both close to the Z axis.  This is just a formal expression.  So, you see that the product is almost 1 minus some small parameter.  And fortunately, resonance, when this something becomes small,  and to understand what happens at this case, we can just look at how spin rotates.  In this case, we have, during the time tau, we have exactly half a rotation around the Z axis.  So, if we started with the electron in state 1, it rotates pi around Z,  then x around the x axis, and then another pi around Z.  So, as a result, we have minus x of the nuclear spin.  If the electron started in a state 0, then we have rotation around x, full rotation around Z, which does nothing,  and then again, rotation around x.  So, our out state is plus x.  That's it. That's a resonance.  And now we can do something good.  But first, we should understand that the resonance is narrow.  Here is just the graph of these functions I showed before.  Timing jitter should be less than 1 nanosecond over 100 microsecond time span.  So, this is very precise requirement should be satisfied to do this.  If we make an error by 10 nanosecond, the fidelity drops to 90% already.  But there is also a beautiful thing that if we're in this resonance, we do nuclear spin conditional rotation.  But if we're exactly integer number of full rotations away from the resonance,  then we have unconditional nuclear spin rotation, which is not interrupted by interaction with the electron.  So, we can do all nuclear gates only by changing the inter-pulse time delay.  And first proof of concept. It first is protected controlled rotation gate.  And here you see if the electron starts in the spin 1, the nucleus oscillates around the x-axis.  But if we are in the state 0, then nucleus stays in place and shows only a little bit of T1 decay.  And this is the state tomography, which shows us that the fidelity of the state, of the bell state created in this way, is 97%.  Although we are about 10 times longer than the electron, the de-coherence time for induction decay time.  Now, we can do something more interesting. T2 star is just a defasing.  We can show that we can go beyond the T2 time and do still high fidelity gates.  So, what's been done here, T2, 50 microsecond, gate time is longer.  What we want to do is to create the bell state.  And again, we can do it with four decoupling pulses on the electron, eight decoupling, 16 decoupling pulses on the electron.  But the only thing we need to care about is keeping precisely the resonant condition.  And when we do it, you see for 16 pulses, we're doing in the same way as if no de-coherence was present here.  So good.  So we have a working gate. We decoupled the electron.  We created the de-coherence protected gate for the nuclear spin.  We're doing it on a time scale which is twice larger than T2.  So let's do some algorithm. Let's... oops. Let's see.  Yes, it works. Good.  So we can take two qubits. We have electron spin and the nuclear spin.  Grover's algorithm converges in one iteration, but still, if we look at the total time needed to perform the algorithm, it is 300 microsecond.  And our T2 time is only 250 microsecond.  So this is the block diagram.  And this is exactly the scheme of the pulses applied in the electron channel and in the nuclear channel.  I just told you we just keep driving very slowly with very small driving.  And we just keep the... adjust the distance between the electron decoupling pulses, either in-perme or half-integer number of rotations.  And yes, if we look at the Grover's algorithm, for instance, if we want our target state is one up, we obtain it with the fidelity 95%.  And here are just the experimental snapshots of different stages of the Grover algorithm.  First, we prepare the equals proposition, flip, do the inversion, flip again, and restore the state.  So the fidelity for the target state one up is 95%.  For the other states, it's 93, 92, 91.  But in all cases, we're above 90%.  Although, again, our algorithm time is 60% longer than the spin echo.  So we can achieve high fidelity beyond coherence, which is great.  So just as a conclusion, the diamond-based quantum information processing becomes competitive,  because not only our work, but in general the community working in this area, demonstrated already spin photon entanglement,  protected gates, and dynamical decoupling.  So we have all three crucial ingredients, at least for small-scale quantum information processing.  Coherence time can be extended. We show 25-fold enhancement.  Other people show comparable figures.  Dynamical decoupling can be efficiently combined with gates.  And we can even do protected algorithms at the times which exceed T2 times, spin echo time, with the fidelity above 90%.  And just as a side note, I think it's nice that this was the first two-qubit computation on individually-addressed state spins.  So thank you for your attention.  Thank you, Slava, for finishing five minutes earlier.  Got us back on schedule, but we do have some time for questions.  So maybe it sounds stupid, but what if you wanted to go beyond T1?  We need to decouple phonons. In principle, it can be done.  But there are other ways to beat that limit.  So just right now it's not the actual limit, because by lowering temperature even to 20K, you have T1 in minutes, a water of minutes.  I'm sorry, I forgot to say the most important thing.  This is all done at room temperature.  I was giving this talk to the mostly diamond audiences.  I forgot to mention the crucial thing. This is all room temperature.  So essentially what you have, you have just an objective, one laser, and a piece of diamond, which you can buy for $40.  And that's it. You have a quantum computer.  No, I'm serious. I'm not joking.  Well, yes, you need microfabrication lab somewhere.  How many carats diamond is this?  Oh, it's about this big, about couple millimeters by couple millimeters.  But don't show it to anybody. It's very ugly.  It looks like a piece of old plastic. It has a lot of nitrogen in it. It's yellow.  This is very nice. So how about you go to like a more than two qubits? What would be the perspective?  More than two qubits?  Well, again, first weekend there is a very successful effort, which recently also there was Nature Physics published last year,  where people were able to place two and well, place maybe.  Okay, there were two NV centers which were close enough to appreciably interact via dipolar interactions.  But at the same time, far enough to be resolved optically.  So there has been already demonstration of two dipolar coupled, individually addressable, and these centers.  Actually, if I had my computer, I even have a little slide on this, because we did the simulations for this.  We also showed that we can go far, far beyond T2 time also.  So the carats protected gates can be done in this case.  For that case, is it also room temperature?  That's limited at T1, T2, T3?  No, no, no, no. I think it was something like 50 kilo. I'm not sure. Let me.  But it can be room temperature as well.  For this system, the difference is in T1, nothing else.  And maybe, let me add one more. There has been spin photon entanglement demonstrated last year.  So just entangling, probabilistic entanglement between two NVs is just a matter of time.  Whoever does it first actually does.  So why do you use this ugly diamond with lots of nitrogen and not a clean one with very few nitrogens?  Well, we use the clean one too. The one with 250 microsecond coherence time, it was a clean one.  But it doesn't really matter. We can do well cases.  Yeah, but if you want to get long coherence time, it's easier.  Yes, we did this. The one with where the algorithm was done, 250 microsecond. It was a clean one.  Well, you still need to decouple carbon-13 spins.  So you cannot go without decoupling anyway.  Thanks again.
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All right, good afternoon. We're ready to start the next session. But before we do just a quick announcement about the banquet, the banquet will be here downstairs where we had lunch on the first day before they moved us outside. And everyone's welcome. So our first speaker is Slavoj Dobrowicki. And he will tell us about quantum dynamics and quantum control of spins and diamonds. And so, thank you very much for coming, for finishing lunch early, coming to my talk. So I will be talking about a few things we have done with our theoretical collaborators, with my former post-doctor Gichlwam, who is now working with Daniel, with the group of Ronald Hansen from Technical University of Delft and from the group of David Arshalom, two hours away from... So recently there have been a lot of interest in individual quantum spins in solid state. And people have studied a lot and some of the systems are probably known to you. These are quantum dots, which have been popular, very popular, for the last two years. Donors and silicon, where they finally achieved the ability to work with a single donor center. And the subject of my talk, and the center in diamond. So why do we care about individual spins? And so let's stay to all four of you, for reasons. Because they're good for at least small to medium scale quantum information processing. They are good for coherent spin electronics and photonics, and this is probably the most fast-growing area of their potential applications. High precision metrology and magnetic sensing at nano-scale. And huge area with very, very impressive potential applications. But of course, as always, we are facing a grand challenge, how to control single quantum spins in solids. And for this, we need to solve quite a few fundamental problems, to understand dynamics of individual quantum spin, to understand how to control it, how to preserve its coherence, and how to generate and preserve entanglement between quantum spins. And for all these, so this is the picture which in the shows what we want to do. We want to be able to pick up a single spin in solid state, do whatever we want, and whatever we want. And spins and diamond, they constitute an excellent test bed for this. They have long coherence time. They are individually addressable and controllable both optically and magnetically. And at our disposal, we have an excellent menu of dynamical decoupling protocols. And just to start with the traditional introduction, usually they are analyzed and classified using Magnus expansion, when we have terms of zero order, first order, second order, and so on, assuming that the distance between the dynamical decoupling pulses is very small. And the simplest one is periodic dynamical decoupling, where when we wait, apply pulse, wait, apply pulse, tau x, tau x. We can symmetrize it, so there has been a lot of introduction. So we can have this CPMG protocol, wait, apply pulse, wait twice, apply pulse, wait. And we can concatenate them, put them inside each other, like this Russian doll, and achieve more impressive protocols. However, for any, probably for any theorist who starts working with experimentalists, there is a moment of truth when you realize that whatever all these fancy and beautiful theories you're doing, they're totally useless, and you have to deal with something much more mundane, something like pulse errors, or the fact that this generator doesn't have enough dynamical range, and all your beautiful suggestions, they either go down the drain, or you have to do something else. So what are the problems with the standard approach? First, we always hear about the norm of certain Hamiltonians. Well, reality, all these norms grow with the size of the bath. So whenever we write this beautiful expression, norm of H multiplied by tau, don't forget that this is essentially infinite. And it is not small parameter, it's actually infinite parameter. Second, validity conditions in reality are not satisfied, because the ultraviolet cutoff is usually too large, but still, well, no DD works. Behavior at long times is not quite clear, because again, of this nature of the cumulant expansion, and the cumulation of pulse errors and imperfections is also not well understood. So this is why we have to usually do some numerical stuff to really help experimentally and assess how the codes will perform in reality. And so we can do it this way or that way, and often we can do much simpler things. They may be not as impressive from point of view, but they produce really good results, such as mintial techniques. So what will be my talk about? First, about quantum control and dynamical decoupling of an e-center. There lately has been spectacular progress. There were three papers published in different journals, probably within the time span of one month or so. At least I remember how Ronald called me at some point and said that Corey put the paper on archive and we have to put our paper on archive, so it was, but it was fun. Second, how to do the coherence protected quantum gates, and I will demonstrate you the coherence protected quantum algorithm in solid state, and correct me if I'm wrong, but as far as I know, this is the first two-qubit quantum computation with individual solid state speed, which we finally achieved. So what are the characters in this story? The simplest impurity in diamond is a substitutional nitrogen. We take away carbon, we put nitrogen instead, we have one excess spin, so there is a spin one half. And these guys, green guys are bad. They constitute our spin bath, our environment, which decahiers us, and they're coupled to each other with long-range dipolar coupling. But when nitrogen meets vacancy, the transformation happens, the bad guy becomes a good guy, we have this beautiful nitrogen vacancy center, which is our central spin system, or our qubit. The red one is the electron spin, which were after, actually it's a spin of two holes, so the total spin is one, and there is also a nuclear spin of nitrogen 14, it will be nuclear spin one or nitrogen 15, spin one half. And they're coupled on-site, and it will play an important role later. And this center is coupled by the polar interaction to a large number of nitrogens. So what's so good about this center? It can be initialized, manipulated, and readout, optically and magnetically. So we can look at a single individual spin in a solid state. Why is it possible there is a very interesting and not even completely understood level diagram, but to make a long story short, we have spin one, we have three sub-levels, 0 plus 1 minus 1. Zero sub-level, if we shine a photon on it, it gets excited to the excited state, stays there for 10 nanoseconds, and falls back and emits a photon. So every time, photon in, photon out, photon in, photon out. But if we're in one of those states, we can be excited, then make what is called in this quantum chemical jargon intersystem crossing, and we get stuck in this shelving state for a long, long time, for 300 nanoseconds, almost infinity. So, and even when we come back, we come back first to m equals zero state. So we started here, but eventually we came back here. And when there is a transition from here to here, no visible photons are emitted. So if we're in m equals zero, photon in, photon out, photon in, photon out. If we're in these two states, photon in, nothing. So, and even when there's nothing over, we're polarized in m equals zero state with typical polarization of order of 90% or higher, which is... So, initialization and redoubt simply by shining a green laser. Nothing else is needed. Now, let's look at the coherence. So here we have this nitrogen vacancy center. We apply some magnetic field. We forget about this level, minus one. It's not of interest for us, it's an idle. And here is our qubit. And we have buff spins, nitrogen atoms. But because of the different parameters, this resonant frequency is very different from the frequency of this transition. So what we're dealing with is a pure defacing. There is just effective level splitting between these two qubit levels. And there is some operator which describes time-dependent action of the bath on the qubit. Purely defacing term. Now, how to deal with this? Of course, if we want to make our... If we want to be completely honest, it's a many-body quantum non-equilibrium problem which we have no way to address analytically, at least unless we have some good parameters, which we don't in this case. But in this particular case, it appears that due to the fact that our bath feels negligible back action from the central spin, we can replace the field created by the bath just as a random C-number field, which is Gaussian stationary and Markovian noise, or the null and back process, which is characterized by two parameters only. Strength of the noise, B, which is determined by the spin-bath coupling between the V and the bath of nitrogens. And the correlation time, which is determined by the coupling between nitrogens, between rings. And we can do direct many-spin modeling. We can run simulations with 20 spins. We can run simulations with 10,000 spins. And again, to make a long story short, we just can confirm that indeed this very simple model is good enough, is adequate for our experiments. Here is the correlation function. Here is what we get for spin echo, just taking numerical simulation and versus analytical model free evolution. So you see agreement is very good. So, qubit in a Gaussian stationary Markovian mean field. Now, just to complete the story, we can look at the experimental Frida coherence. These red dots are experiment, gray line theory. We can extract C2 star time, and therefore we can extract the coupling of the qubit to the environment. Here you see modulation induction signal. It is because of hyperfine coupling of the nuclear spin. Next, just to make sure that our mean field model is indeed correct, we can do spin echo. And we see that it indeed decays as theory predict, e2 minus t cube decay. And the correlation time of the both is 25 microseconds, which is long enough for us to do dynamical decoupling. So, our first step, quantum control and dynamical decoupling. Extending coherence time of a single MV center. First, of course, since nobody has done it before, on MV centers before us, the question was always which protocol to choose from this plethora of protocols. And we started analyzing one after another. First, we started with PDD, like simplest protocol, the most obvious choice. And we ran into the problem right away. In short times, it decays four times faster than we would like to. Only at longer times, it becomes here instead of four thirds, we have one third. So, it's good, it's simple, but unfortunately it decays fast initially. Not what we want. So, let's try CPMG. It appears that CPMG has this slow decay at all times. Beautiful. Now, of course, we want to try to concatenate it. And we see that fast decay at all times, if we concatenate PDDs, and if we concatenate CPMGs, slow decay at all times, and no improvement. Absolutely no improvement, which means that we will be just programming a more difficult sequence without any actual, any advantage. So, why? Well, first of all, it's kind of an obvious statement, but I want to reiterate it, that coherence time can be extended well beyond 25 microseconds, well beyond the correlation time of the bus. Second, Magnus expansion gives us completely wrong predictions, because instead of this decay for PDD or for CPMG, we have, instead of tau to the fourth, we have tau to the cube. Symmetrization or concatenation gives no improvement, and simply because we have Orson-Nullenbeck noise, with a heavy tail, one over omega squared. Second moment of this noise is formally infinite, and it corresponds to what I started with, that the norm of the butt Hamiltonian is infinite. Actually, it's not quite infinite, but it's at 5 gigahertz, and it's much faster than anything, any other timescale in our experiments. So, we know the protocol, CPMG. We start running it, and we immediately see that X component is protected very well, and Y component is not protected very well. So, what's wrong? Because also, obvious control pulses are not perfect. And it's actually interesting to see why they are not perfect. When we're trying to control our envy center with the actual experimental situation, you see that during the whole pulse, we have only one, two, three oscillations, and we have a ring down. So, again, all these beautiful rotation frame approximation, inapplicable. If we have very slow driving, then we indeed see what we expect, to see beautiful rib oscillations of the spin. But when the driving becomes stronger, because this is what we need to achieve very short pulses, you see that we don't have the, you know, the same kind of oscillations, but we have a very short pulse, and we have a very short pulse, and we achieve very short pulses. You see that we don't have these beautiful nice oscillations. Instead, we have kind of spin stalls in some states, then quickly jumps from up to down, stays there for a while, jumps halfway, then stays, and shows the sequence of past jumps and stalls. So, and this is not an experimental noise, as you see it's reproduced completely by simulation. This is just what happens when we try to drive our spin very fast. And of course, it introduces a lot of errors in the pulses, and we need to characterize them, so we did some work how to develop the scheme for character of the pulse errors. Then we should study how the errors accumulate. We also did that, and we saw some interesting stuff. I hope Kavi will talk more about this. So, again, bottom line is that if we apply pulses along X, X component is preserved well, why not? But if we apply pulses along X and Y, and use these X, Y, 4 sequence, which was introduced yesterday, we see that X and Y are protected well, and coherence time is indeed extended. So, here you see that the coherence time is in the region of about 7 milliseconds, although we started with 2. Good. We can look at other a pyrotic sequences, UDD, QDD. So, what do we see? Well, first we do not expect them to work very well, because we have no hard cutoff in the buff, and indeed they do not perform as well as we expected. But there is a deeper problem with them if we look at the robustness to the pulse errors. And here are the simulations with imperfect pulses, whether the parameters of every pulse are taken from experiment. So, as you see, still X, Y, 4 performs much better than QDD, and of course incomparably better than UDD. So, robustness to pulses plays an important role. But, again, if we choose our protocol correctly, if we take into account the errors, and if we do things clear, in a clean manner, then we see exactly what theory predicts. And what does it predict? First, for any number of pulses, the decay should have this form, E to minus T cubed. So, if we take spin echo, 4 pulses, 8 pulses, 136 pulses, and rescale them, they should fall on the same master curve, and this is exactly what you see. And if we measure the coherence time as a function of the number of pulses, it should demonstrate N to the 2 thirds. And, again, this is exactly what experiment shows. So, applying 136 pulses, coherence is increased by a factor of almost 30, we achieve 90 microsecond, which is great. And, well, actually, I wrote here no limit inside, we're limited at this time scale, we started being limited by T1 processes. So, we're practically pushed dynamical decoupling to the limit. But this is what happened last year. Since then, we're doing some other things, using dynamical decoupling to improve single-spin magnetometry, or to do the detailed probing of the mesoscopic spin bath, which surrounds a single NV center. So, we take a single center and study around 10 spins, which form an effective bath for these particular centers. So, essentially, again, it gives us ability to study mesoscopic spin dynamics. But this is only half of the story. Protecting spins is good, but we want to do something useful with them. And for this, we need many NVs to talk to each other, to process information in some way. And there were one particularly useful ways to make NVs talk to each other, is to use hybrid systems. Inject systems are very popular now, because they combine best features of different qubits, which we know about. So, here in particular, there are suggestions to put NVs on conciliators and use the nanomechanical oscillators as data bus between the NV centers. There's been a suggestion to use spin chains, but the simplest one, which you can think of, and the most probably widespread, is when we use electron spin as a processor for quantum information, and nuclear spin as a memory. So, anyway, we have many spins, and they decay here. We want to do something useful with them. So, if we think about standard mode of quantum operation, of course, it doesn't work. What we can do is unprotected quantum gate, but then again protected. Or we can do protected storage, but then we kill the coupling, not only between the different qubits, which have different time scales, but also between our qubit and the bus. So, we can efficiently preserve the state. We can efficiently isolate our NV center. But in order to perform some interesting computation, we need to change the state. We need to keep electron spin, for instance, talking to the nuclear spin. This is what we want to achieve. And to do this, we need to be able to combine dynamical decoupling pulses with the gate operation in the seamless fashion. So, decouple and compute at the same time. And again, since we are dealing with hybrid system, different qubits have very different coherence and control time scales. So, our electron, we start decoupling our electron, nuclear spin just barely starts to move. So, by the time the electron spins, it occurs, nuclear spin just starts to rotate. If we... So, if we try to control the nuclear spin, it takes much longer than the electron decoherence time. So, we have essentially the choice. Either decouple the electron, apply dynamical decoupling, but then no gates possible because we decouple it from the nucleus. Or we can do gating without dynamical decoupling. But then again, we don't have any gates. So, there is... Of course, I wouldn't talk about this if there was no way out. And the way out is to use the resonance in the system. This... What... I don't know, for this talk, I called it guard. Gates with resonance decoupling. So, cut our system in more detail. In the rotating frame, this is the Hamiltonian which describes it. SZ is a very fast, decayed electron spin. I is the very, very slow, barely moving nuclear spin. So, you can see, for instance, that here we have 2 MHz hyperfine coupling, but the driving we apply to the nuclear spin is only 18 kHz. And this is a very typical figure. So, let's look what happens in the rotating frame. If the electron spin is on a state 1, SZ equals 1, and we have nuclear rotating around Z, because this term kills that one. If the electron is on a state 0, this term is absent, we have only driving. The nucleus slowly, slowly rotates around the X axis. So, the only problem here is that the electron switches very fast between Z. This is the essence of dynamical decoupling. And... But the nuclear spin should keep track of what was the electron state long before we started decoupling. At first sight, it's kind of contradictory, but in fact, if we look at the motion of the nuclear spin, if we start from the state 0, it will be 0, pulse on the electron spin, so the electron spin will be 0, 1, 1, 0. If we start at the electron spin on state 1, it will be state 1, 0, 0, 1. And in general, the Hamiltonians are different, but these rotation axes for the nuclear spin, they are closed. They are very close to each other, simply because they are both close to the Z axis. This is just a formal expression. So, you see that the product is almost 1 minus some small parameter. And fortunately, resonance, when this something becomes small, and to understand what happens at this case, we can just look at how spin rotates. In this case, we have, during the time tau, we have exactly half a rotation around the Z axis. So, if we started with the electron in state 1, it rotates pi around Z, then x around the x axis, and then another pi around Z. So, as a result, we have minus x of the nuclear spin. If the electron started in a state 0, then we have rotation around x, full rotation around Z, which does nothing, and then again, rotation around x. So, our out state is plus x. That's it. That's a resonance. And now we can do something good. But first, we should understand that the resonance is narrow. Here is just the graph of these functions I showed before. Timing jitter should be less than 1 nanosecond over 100 microsecond time span. So, this is very precise requirement should be satisfied to do this. If we make an error by 10 nanosecond, the fidelity drops to 90% already. But there is also a beautiful thing that if we're in this resonance, we do nuclear spin conditional rotation. But if we're exactly integer number of full rotations away from the resonance, then we have unconditional nuclear spin rotation, which is not interrupted by interaction with the electron. So, we can do all nuclear gates only by changing the inter-pulse time delay. And first proof of concept. It first is protected controlled rotation gate. And here you see if the electron starts in the spin 1, the nucleus oscillates around the x-axis. But if we are in the state 0, then nucleus stays in place and shows only a little bit of T1 decay. And this is the state tomography, which shows us that the fidelity of the state, of the bell state created in this way, is 97%. Although we are about 10 times longer than the electron, the de-coherence time for induction decay time. Now, we can do something more interesting. T2 star is just a defasing. We can show that we can go beyond the T2 time and do still high fidelity gates. So, what's been done here, T2, 50 microsecond, gate time is longer. What we want to do is to create the bell state. And again, we can do it with four decoupling pulses on the electron, eight decoupling, 16 decoupling pulses on the electron. But the only thing we need to care about is keeping precisely the resonant condition. And when we do it, you see for 16 pulses, we're doing in the same way as if no de-coherence was present here. So good. So we have a working gate. We decoupled the electron. We created the de-coherence protected gate for the nuclear spin. We're doing it on a time scale which is twice larger than T2. So let's do some algorithm. Let's... oops. Let's see. Yes, it works. Good. So we can take two qubits. We have electron spin and the nuclear spin. Grover's algorithm converges in one iteration, but still, if we look at the total time needed to perform the algorithm, it is 300 microsecond. And our T2 time is only 250 microsecond. So this is the block diagram. And this is exactly the scheme of the pulses applied in the electron channel and in the nuclear channel. I just told you we just keep driving very slowly with very small driving. And we just keep the... adjust the distance between the electron decoupling pulses, either in-perme or half-integer number of rotations. And yes, if we look at the Grover's algorithm, for instance, if we want our target state is one up, we obtain it with the fidelity 95%. And here are just the experimental snapshots of different stages of the Grover algorithm. First, we prepare the equals proposition, flip, do the inversion, flip again, and restore the state. So the fidelity for the target state one up is 95%. For the other states, it's 93, 92, 91. But in all cases, we're above 90%. Although, again, our algorithm time is 60% longer than the spin echo. So we can achieve high fidelity beyond coherence, which is great. So just as a conclusion, the diamond-based quantum information processing becomes competitive, because not only our work, but in general the community working in this area, demonstrated already spin photon entanglement, protected gates, and dynamical decoupling. So we have all three crucial ingredients, at least for small-scale quantum information processing. Coherence time can be extended. We show 25-fold enhancement. Other people show comparable figures. Dynamical decoupling can be efficiently combined with gates. And we can even do protected algorithms at the times which exceed T2 times, spin echo time, with the fidelity above 90%. And just as a side note, I think it's nice that this was the first two-qubit computation on individually-addressed state spins. So thank you for your attention. Thank you, Slava, for finishing five minutes earlier. Got us back on schedule, but we do have some time for questions. So maybe it sounds stupid, but what if you wanted to go beyond T1? We need to decouple phonons. In principle, it can be done. But there are other ways to beat that limit. So just right now it's not the actual limit, because by lowering temperature even to 20K, you have T1 in minutes, a water of minutes. I'm sorry, I forgot to say the most important thing. This is all done at room temperature. I was giving this talk to the mostly diamond audiences. I forgot to mention the crucial thing. This is all room temperature. So essentially what you have, you have just an objective, one laser, and a piece of diamond, which you can buy for $40. And that's it. You have a quantum computer. No, I'm serious. I'm not joking. Well, yes, you need microfabrication lab somewhere. How many carats diamond is this? Oh, it's about this big, about couple millimeters by couple millimeters. But don't show it to anybody. It's very ugly. It looks like a piece of old plastic. It has a lot of nitrogen in it. It's yellow. This is very nice. So how about you go to like a more than two qubits? What would be the perspective? More than two qubits? Well, again, first weekend there is a very successful effort, which recently also there was Nature Physics published last year, where people were able to place two and well, place maybe. Okay, there were two NV centers which were close enough to appreciably interact via dipolar interactions. But at the same time, far enough to be resolved optically. So there has been already demonstration of two dipolar coupled, individually addressable, and these centers. Actually, if I had my computer, I even have a little slide on this, because we did the simulations for this. We also showed that we can go far, far beyond T2 time also. So the carats protected gates can be done in this case. For that case, is it also room temperature? That's limited at T1, T2, T3? No, no, no, no. I think it was something like 50 kilo. I'm not sure. Let me. But it can be room temperature as well. For this system, the difference is in T1, nothing else. And maybe, let me add one more. There has been spin photon entanglement demonstrated last year. So just entangling, probabilistic entanglement between two NVs is just a matter of time. Whoever does it first actually does. So why do you use this ugly diamond with lots of nitrogen and not a clean one with very few nitrogens? Well, we use the clean one too. The one with 250 microsecond coherence time, it was a clean one. But it doesn't really matter. We can do well cases. Yeah, but if you want to get long coherence time, it's easier. Yes, we did this. The one with where the algorithm was done, 250 microsecond. It was a clean one. Well, you still need to decouple carbon-13 spins. So you cannot go without decoupling anyway. Thanks again.
quantum dynamics and quantum control of spins and recently there have been a lot of interest in individual quantum spins in solid state. And potential applications. High precision metrology with very, very impressive potential understand dynamics of individual quantum spin, to understand how to control it, how to preserve for this. They have long coherence time. They are protected quantum gates, and I will demonstrate characters in this story? The simplest impurity in diamond is a substitutional nitrogen. We take it will play an important role later. And this manipulated, and readout, optically and magnetically. So we can look at a single taking numerical simulation and versus analytical story, we can look at the experimental Frida And the correlation time of the both is 25 control and dynamical decoupling. Extending measure the coherence time as a function of the bath for these particular centers. So, between the NV centers. There's been a suggestion you can think of, and the most probably But in order to perform some interesting combine dynamical decoupling pulses with the gate operation in the seamless fashion. So, decouple are dealing with hybrid system, different qubits have very different coherence and control time we start from the state 0, it will be 0, pulse on times longer than the electron, the de-coherence decoupling pulses on the electron, eight longer than the spin echo. So we can achieve high fidelity beyond coherence, which is great. So small-scale quantum information processing. figures. Dynamical decoupling can be efficiently