Robust Dynamical Decoupling
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Transcript: English(auto-generated)
00:00
From Dortmund, we'll be talking about robust dynamical decoupling. Yes, thank you. I think you can hear me. So it's a pleasure to be here, my first QEC conference. And I would like to start with introducing the people who did most of the work I'm going to talk about.
00:20
One of them is here, Gonzalo. Actually, he's talking later in the same session. And Alexandre Martins de Sousa is not here, but he also contributed a lot to the work. And I hope I will have time to talk also a little bit about these two people or their work. Mark Olovarich is a graduate student
00:41
working on optical storage. And Jonghyun Shim, who's working on diamond and v-centers. So what I would like to discuss today is decoherence and the fight against decoherence by dynamical decoupling. So what we have, we have a whole qubit.
01:01
But unfortunately, that's coupled to its environment. And that leads to decay of the coherence. And of course, we would like to do something against that. In other words, we would like to suppress this coupling and hopefully achieve some long-lived coherence. And that's what I would like to discuss in a number of different settings
01:20
and hopefully convince you that this can actually be done in a real experiment. So I will start with some experimental aspects of dynamical decoupling, introduce some robust sequences using, for instance, symmetry in time. And then if I still have time, discuss some applications to single spins
01:44
and to storage of photons in solid. So what I'm going to consider here are systems which are subject to dephasing, phase errors. Now, this can be pictured as by the image you've seen,
02:03
for instance, in Daniel's talk, this dephasing. And you can rephrase that by applying a refocusing pulse, which changes the orientation. And if they process in the same way, they will eventually come back to the original location,
02:23
create an echo, and undo the dephasing. That's, of course, what we want. And that's what was first achieved by this experiment, which also Daniel showed you before, the Hahn echo. Obviously, this is a time asymmetric version
02:41
of the same person. We will do most or most of the experiments that I will discuss have been done in this sort of lab rat, as we like to consider it. It's a molecular crystal called adamantine, or the molecule is called adamantine. Cubits are carbon-13 nuclear spins.
03:03
And they are essentially individual nuclear spins, because in each molecule, you will mostly see zero or one of those carbon-13 spins. But they see an environment full of protons,
03:21
1H nuclear spins. And these form a spin bath, a fluctuating spin bath, which actually then causes the dephasing. The Hamiltonian that we have in this case can be written as an Ising Hamiltonian, summed over all the protons in the sample. And the fact that this is really a good approximation
03:44
to pure dephasing is shown by the ratio of T1, longitudinal relaxation time, versus dephasing time, which is about 10,000 in this molecule. So when you create coherence in it, just let it evolve.
04:01
Then you see a free induction decay, which vanishes on a time scale of a few hundred microseconds. Obviously, that's pretty short. Now, if you try Hahn's solution to that, you create a Hahn echo. And indeed, you can extend the lifetime of the system, unfortunately not too long, by about a factor of two
04:21
or slightly more than two. So what's the problem? Why does it not live longer than that? And the reason is that these protons in the environment, they're not static. They create dephasing rate, which fluctuates in time. And that's what we have to fight.
04:41
And the solution against that is that you don't apply just a single pulse, but you apply a sequence of pi pulses. And each of those pi pulses inverts this heteronuclear coupling. So you go from plus, minus, to plus, to minus, to plus,
05:00
and so on. And if you average over all these Hamiltonians, of course, you get an average Hamiltonian, which is zero in the lowest order. The condition for that to happen is that the delay between the pulses should be short enough, short on the correlation time of the bath.
05:20
In other words, the next pulse has to come before the environment has had time to change significantly. And there's a lot that can go into this timing of the sequence. You've heard about that in Mike Bjorkzuk's talk and in Neil Davidson's talk yesterday. You will hear much more about that in Gonzalo Alvarez' talk.
05:43
It's interesting. It can be useful. You can adjust the timing, for instance, by Eurex formula to improve the decoupling efficiency. Or you can use it to learn something about the environment, as Gonzalo will show you. This is not what I'm going to talk about.
06:01
I will keep all the timing fixed. I will have identical delays between all the pulses. I'm going to talk about something different. But let me first state what I want to achieve here, what I want to show you. Our goal is to keep the coherence of the qubits alive for as long as possible with finite resources.
06:24
That's all we have available. We are poor experimentalists. We can't have infinite amplitudes. And unfortunately, our controls are not perfect, as you will see. So let's see what you can do. This is the atom-antane molecule.
06:41
This was the free induction decay I showed you before. Now on a logarithmic scale, that was the Hahn echo. Now let's see what happens if you apply not the single echo pulse, but the sequence of echo pulses. And you see, it survives longer. The tau here is the delay of the pulses. And you see, if we make this time shorter,
07:01
the survival of the coherence gets longer. So applying more pulses obviously helps. That's perhaps not too surprising. But let me summarize that this was the survival time of the free induction decay. The Hahn echo brought an improvement of about the factor of 2. And the more pulses we apply, the longer the relaxation
07:22
or defacing or decoherence time gets. This is almost three orders of magnitude improvement, which is pretty nice. So applying more pulses helps. Unfortunately, that's not the end of the story. You cannot continue forever. There is a limit. And in this regime, apparently, applying more pulses hurts.
07:41
So it seems here, the pulses actually destroy the coherence rather than preserving it. And we'll see if we can do something against that. But first, let me discuss another issue. This is related to my question yesterday or after near Davidson's talk. This measurement was all done with what
08:01
we call the longitudinal initial condition. That means the initial preparation orients the spins in the direction of the rotation axis of these refocusing pulses. And unfortunately, you cannot always do that. I mean, in a quantum computing environment, you don't know what the state is.
08:21
And so sometimes, or in most cases, it will not be parallel to the rotation axis. So you also have to look at the different case, what we call transverse initial condition. This is the rotation axis. And this is now the orientation of the initial magnetization. And you see, in this case, if you apply more pulses,
08:41
it gets worse. The coherence time does not increase. It decreases. And that's because in this regime, the pulse imperfections, as I said, we're poor experimentalists. We cannot apply perfect pulses. They destroy the coherence. So let's see what we can do. And that's why I was so surprised that near
09:01
had a homogeneous decay or isotropic decay, almost isotropic decay. So here you see orders of magnitude difference. And as I said, this is due to the pulse imperfections. And let's see what we can do against that. Let me start with a very simple example. The simplest cycle of the CPMG sequence
09:22
is just two pipe pulses. And I've arbitrarily chosen the y-axis as the direction of the rotation. So if you have two ideal pipe pulses, obviously you get the unit operator apart from the overall phase. Now, as I said, this is not what we do in the lab. But what we do in the lab, we always apply a pipe pulse plus some additional angle delta,
09:43
which can be positive or negative. This is just the experimental imperfection because the amplifier has noise, for instance. And the radio frequency coil has in homogeneous distribution of the field and so on. So if you apply two such pulses,
10:00
obviously you don't get the unit operator anymore. But you get now a rotation around the y-axis by an angle 2 delta. Why didn't that hurt us in the first place? Well, if we have the initial condition along the y-axis, then the density operator commutes with this overall propagator here. And it has no effect on the state.
10:23
But if we now choose the transverse initial condition, the propagator does not commute with the density operator anymore. And the errors of all the pulses actually add in this case. And that's why we destroyed the coherence in this time. And the solution to that is very simple. Actually, instead of rotating both times
10:40
around the same axis, you just invert the rotation axis. So you combine pi plus delta with minus pi plus delta. And the overall rotation is again 0. That's what you want. We call this sequence now CPMG2. And you see here how it works. This was the transverse initial condition
11:00
I showed you before. That's the decay. After a few dozen pulses, you have no magnetization left. But if you alternate the rotation axis, you get orders of magnitude improvement. You get, again, long-lived coherence time. Still, it's not completely isotropic. It's asymmetric because you always rotate around the y-axis or minus y-axis in this case.
11:24
But something that we can learn from that is that it's useful to not rotate always around the same axis because that's a sure recipe that the errors add. What you want is you combine different rotation axes. And that's exactly what I'm going to talk about now.
11:43
But let me state again what we want to do and what the problem is. In principle, if you write down dynamical decoupling, you have short pulses. And short delaying them, you can get perfect refocusing. Now unfortunately, we can't do that.
12:01
But pulses that we have available, they have finite strength, which means they have finite duration. They have flip angle errors, as I showed you. Also, you cannot always be exactly on resonance for all the qubits in your system. And in some cases, we have errors which we don't even understand. We see that we don't have perfect rotations.
12:22
We don't know exactly what they are. And that makes it, of course, hard to compensate. And what I'm trying to show you as a possible solution is that you use pulses which are insensitive to these experimental imperfections, or you combine pulses
12:41
to robust sequences, robust dynamical decoupling sequences. Or if I may rephrase what you said, you use imperfect pulses to simulate perfect dynamical decoupling sequence. I think that was about what you suggested. Now, this was actually recognized many years ago
13:05
in a completely different environment. Maudsley looked at dynamical decoupling or refocusing in magnetic resonance imaging. And the problem that he had was very similar.
13:20
He could not make sure that the initial condition was aligned with the rotation axis. And he saw that if it was aligned, he had perfect refocusing. But if it was out of phase, then the magnetization decayed very quickly. And the solution which he came up was, instead of using a single rotation axis, he just alternated the rotation axis
13:40
between the x and the y-axis. And he got very good performance independent of the initial condition. And there were several subsequent papers which improved the sequence. But let's see how that works. This is the performance of the CPMG sequence. In other words, each pipe pulse has the same rotation axis
14:02
for different flip angle errors. And we calculate the fidelity after 20 pulses. 20 pulses is a fairly short sequence. But you see, with the CPMG sequence, you completely lose the signal, even if your flip angle error is only 1% or a few percent.
14:20
So definitely not what you want. And if you compare that with the XY4 sequence, you see that you can get good performance over a much wider range of experimental imperfections. And I'm going to show you a few other sequences. For instance, in our hands, the best sequence was this KDD sequence, which I'm going to talk about later.
14:45
But let's consider it in a slightly wider context. Flip angle errors are, under our conditions, probably the most important source of error. But it's not the only one. And in many cases, you have to consider multiple possible errors.
15:01
And that's what I'm doing here. Here we have two main errors. One is the flip angle error along the horizontal axis. And along the vertical axis, I have an offset error. In other words, you cannot apply the pulses exactly at the resonance frequency of the qubits. And what happens for the CPMG sequence
15:22
is that you very quickly lose the coherence. What I plotted here is color-coded the fidelity after 100 pi pulses. And you see, again, very small flip angle errors means you lose your signal. You can improve that, as I said, for instance, with the XY4 or PDD or CDD1 sequence.
15:42
And you see this excellent improvement in terms of the flip angle. It's less impressive in terms of the offset in the other direction. Because it's CDD1, you can, of course, iterate. You can do CDD2. And you get, again, further improvement.
16:01
And actually, this is already pretty good. If you consider here the highest contour level, that means 99.9% fidelity after 100 pulses. So this is roughly 10 to the minus 5 error for a single pulse. So that looks pretty good. But of course, we would like to do better.
16:21
And one approach for that is called composite pulses or robust pulses or compensated pulses, which were introduced in NMR by Levitin Freeman in 1979. And I would say that's pretty similar to what you heard about from Lorenz yesterday.
16:40
Was it one day? One day, sorry. So the idea is that instead of using simple pipe pulses, we use robust pipe pulses and insert them, for instance, into this X sequence. Replace the simple rectangular pipe pulses by composite or compensated pipe pulses. Now, this is the pipe pulse that or compensated pipe pulse
17:05
that Levitin Freeman introduced. This is, by the way, how it works. You want to go from the plus side to the minus z-axis. And these trajectories here correspond to different flip angle errors. And the idea of this my pulse in the middle is that it takes you from above the equator
17:22
to down to the equator. And then you end up close to the south pole, no matter how good your RF flip angle is. Now, this is the earliest one. But there have been improvements in the meantime. And for us, the best pulse to replace the simple pipe pulse
17:41
turned out to be this sequence of five pipe pulses. For a number of reasons, I'm not going to discuss about it. In the literature, it's sometimes called the Knilpulse. The earliest reference we found to it was in this 1985 paper by Rob Teeko, Pines, and Guchenheimer. So you have five pipe pulses.
18:00
And each of them has a different phase, 60 degrees, 0 degrees, 90 degrees, 0, 60 degrees. So we take this pipe pulse and insert it into the sequences that I showed you before. If you use the CPMG sequence and just replace each pipe pulse by one of these Knilpulses, then you get an incredible improvement.
18:23
I mean, from virtually nothing here, you get a pretty large area here where the sequence performs quite well. You can also insert it in the XY4 sequence, of course. And this looks now almost perfect. I mean, all this area here, you have a fidelity of 99.9%
18:41
after 100 pipe pulses. And that's nice. There is one drawback. Because we replaced each pipe pulse by five pipe pulses means we actually increased the power deposition by a factor of five. And the question is, can we avoid that?
19:01
And of course, I wouldn't ask the question if I didn't have a solution to it. And the idea is very simple, actually. I mean, what we replaced was a pipe pulse and a delay by five pipe pulses and delay. And obviously, that's not the smartest solution. But what you can do is you split these five pipe pulses
19:20
up and have a fifth of the delay in between. So you distributed pulses and delays homogeneously throughout the sequence where you want to decouple. Now, this sequence of five pipe pulses replaces the single pipe pulse. And you can insert it again into the XY4 sequence. So you have these five pulses for the X pulse
19:42
and these five pipe pulses for the Y pulse. And together, this gives you another decoupling cycle. Actually, you have to repeat it twice to get the reals. And this we call then KDD. That's the sequence I showed you before in the one-dimensional plot. And just let's take a look at it in the 2D plot.
20:02
This was what I showed you before. And KDD performs pretty similar to the compensated XY4 sequence. But the duty cycle here is five times lower than here. It's the same duty cycle as above here. So this is clearly a very nice improvement.
20:21
But so far, I've only shown you theoretical results. These were simulations. We only looked at the effect of pulse imperfections. There was no environment. So in other words, we don't have decoupling yet. So we have to see if we can actually decouple with these pulses. And we can simulate that.
20:41
But I'd prefer to show you experimental results here. These were the simple CDD sequences, CDD1, CDD2. And you see in each, you get improvement when you apply more pulses. But there is an optimum. And then the pulse imperfections take over. Now, if you replace the pulses in these CDD sequences
21:02
by robust pipe pulses, you don't get this saturation anymore. But the coherence time keeps improving. And we go here essentially to duty cycle one. Duty cycle one means you apply only pulses, no delays in between. Duty cycle is the total duration of the pulses divided by the total duration of the cycle.
21:24
And in principle, it's important to have good performance for every duty cycle. I mean, high duty cycle would be a typical application where you want to store information, say quantum memory. You don't want to do anything.
21:41
You want to implement the unit operator. You can apply as much power as you want. But if you want to compute, for instance, you may not be able to keep seeing all the time. And then you have to lower your duty cycle. So in principle, it would be good to have something which works over the whole range of duty cycles.
22:02
And that's not yet the case. You see here, the simple pulse sequences perform better than the robust sequences. And why is that? Well, we increase the duty cycle by a factor of 5 here. And that's exactly what happened here. But if we now take the KDD sequence, you see it performs well over the whole range.
22:23
And actually, at least for this application, it performed best over the whole range of possible applications. So this turned out to be a quite useful way to improve the decoupling performance
22:42
and get a sequence which turned out to be quite robust under our conditions. Now I will discuss a few other ways to improve decoupling performance. And always, the main focus will be that we don't want to create a lot of overhead.
23:01
I mean, of course, you can apply harder pulses, put in more power. But that's actually not what you want. And in one way, to reduce pulse imperfections, or the simplest way maybe to reduce pulse imperfections, is just not to apply any pulses.
23:21
So my PhD supervisor used to say, every pulse is one too many. And in a way, that's right. I mean, the best pulses, or the least errors, are the pulses which you don't apply. And I would like to show you one situation where this is possible.
23:40
We call that virtual pulse. So this is the CDD scheme. You start with the XY4 sequence. And these are the first two pulses of the XY sequence. And you insert the XY4 sequence into the delays between the pulses. Now, with virtual pulses, I mean that these pulses here,
24:02
we don't really apply. We essentially let the pulses act on the pulses that we insert. And that means instead of having XY, XY here, we have X minus Y, X minus Y. And then instead of applying a Y pulse here, you invert the X pulse, and so on.
24:23
And we call that VCCD for virtual concatenated dynamical decoupling. Let's see how that works. This is the CDD2 sequence. Again, in a two-dimensional representation, each point is an experiment where we measure the survival probability after 100 pulses
24:44
as a function of two experimental parameters. One is the offset here, and one is between the pulses. And what you see here, if you apply your pulses on resonance, you get a pretty good performance. Eventually, of course, the magnetization decays.
25:00
But for short pulses, spacings, and small RF offset, it works well. But even for small RF offsets, you lose the magnetization. And if you now use these virtual pulses instead of the real pulses, you get a significant improvement. We have a much broader range of offsets
25:22
over which the magnetization survives. So we have improved the performance in this case with, well, actually negative overhead. This is a comparison with the KDD sequence, which looks similar to the VCCD2 sequence. And of course, you can extend that.
25:41
So we have achieved here an improvement of the performance with negative overhead. So we put in less power and get performance. And well, you cannot always do that. But sometimes, at least, you can improve the performance
26:00
without putting any overhead in. That's what I'm going to discuss now. If you look at literature on the XY4 or PDD or CDD1 sequence, you see actually two versions. The original one looks roughly like this. And this is the one which you normally find in the quantum computing literature.
26:22
It's always the same sequence, x, y, x, y, with delays tau in between. The difference is that in this old paper, the pulses are placed symmetrically within one cycle. So you start with a tau over 2 delay and end with a tau over 2 delay.
26:41
The other version, you have a tau delay first, and you end with a pulse. Of course, that doesn't matter if you only look at the average Hamiltonian of the sequence. But it does matter if you look at higher orders. And it does matter if you look at the experiment. Actually, if you measure the echoes,
27:00
echo sequence in this version, you see twice as many echoes as in the lower version. Why? Well, you start with a signal here. It decays and refaces in the window between the pulses. In each window, you get an echo. Here, it defaces for a time tau and refaces after 2 tau.
27:21
That's why you get half as many echoes. And that already shows you some problem with this sequence. Basically, here, the environment has twice as much time to change as in the upper sequence. And well, this is the experimental comparison.
27:41
The other was just a plot. This is with the symmetric version. You see the echoes between each pulse. And if you apply the AC version, you get half as many echoes. Now, this is just for a single cycle with two pulses. You see that the decay is slightly faster for the asymmetric version than for the symmetric version.
28:03
It's a small improvement. I certainly don't want to say that this is a big improvement. But there is zero overhead, which seems worthwhile to take. Now, it also matters if you start to look for better cycles which are based on that.
28:25
And the idea is that you want to have a cycle which is time reversal symmetric. And what that means is if you look at the tockling frame Hamiltonian between the pulses, and then you compare the sequence of tockling frame
28:41
Hamiltonians forward in time with the opposite direction, then a time reversal symmetry means that you get the same sequence. And the advantage of that, if you have such a sequence, is that all ordered average Hamiltonian terms for a time symmetric sequence vanish.
29:02
So again, you have something for free. There is no overhead associated with that. And that's helpful. Let's see what that means. There is another. I mean, that is one way of saying
29:20
why you have a slightly longer survival time in the symmetric sequence that I showed you before than in the asymmetric sequence. But there is an important corollary if you now start to combine such cycles. If you calculate the average Hamiltonian of such a sequence and you start with a system plus system environment
29:41
direction plus environment Hamiltonian, what you get as the average Hamiltonian in both cases, the load is the environment Hamiltonian. That's what you want. But in practice, you get additional terms, which can be higher order contributions, which can be error experimental imperfections. And all these error terms hurt.
30:01
And that's why the asymmetric sequence has more error terms there so it decays more rapidly. Now, every sequence can be improved. And one way to improve this xy4 sequence is to invert it in time. So you go from xyxy to yxyx.
30:22
And then you put them behind each other. So you get the new cycle, which is twice as long, and which is, of course, inherently now time symmetric. You can also do that with the asymmetric version. And you get now also time symmetric sequence. However, you started with different sequences.
30:41
And that can also hurt in the combined sequence. And for instance, you can combine two of these xy8 sequences to an xy16 sequence by combining the xy8 sequence with its Hermitian conjugate. So OK, I have to get a bit faster, it seems.
31:03
This is a result of experimental quantum process tomography. You see for the xy4 sequence, the asymmetric and the symmetric perform almost identically. But if you combine them to the xy16 sequence, here is the asymmetric version and here is the symmetric version.
31:21
You see that obviously the asymmetric version has an additional error term, which causes you a precession of the magnetization, which destroys the fidelity much faster. So you can also implement these or use these symmetrized versions for the CDD sequence.
31:41
In the normal CDD case, you insert xy4 into the gaps in such a way pulses here are back to back. Then you can eliminate, for instance, double y pulses if you want. And you can also do that for the symmetrized version. And you then insert xy sequences in a symmetrized,
32:02
so you have delays more equally distributed. And at least in our hands, that again, perform better if you use the symmetrized version. You see that the symmetric CDD2 sequence performs somewhat better than the asymmetric CDD2 sequence. Again, it's a small improvement, but there is no cost.
32:24
And that seems to be worthwhile. Now, I have a few minutes left. Just let me show a few applications of that, which are now not on nuclear spins in solids, but on different spins. In this case, it's an electron spin in diamond.
32:41
You know these systems when you have the free induction decay. It vanishes tip time scale of less than a microsecond. You can increase that with a Hane code to a few tens of microseconds, or you can apply CPMG sequence to some hundreds of microseconds. Now, in this case, we also saw a significant difference
33:02
for the CPMG sequence, depending whether we choose the longitudinal or the transverse initial condition. Here after 20 pulses, we have very little signal left. And if you look at the function of the angle orientation of the initial condition, you see that this clearly gets worse.
33:20
So again, we look for another sequence which performs better independent of the initial condition. And of course, we've already found that we used KDD2. KDD2 means two cycles of KDD, which is 40 pulses. And you see, we get a pretty independent performance.
33:41
And you can then apply more pulses. You increase the analysis, and you get longer T2 times, actually up to a few milliseconds in this case, which is quite nice. And it scales roughly with the 2 third power of the number of pulses, which
34:00
is what you expect if you have a Lorentzian spectral width. And sorry, that was a bit too quick. We get the few milliseconds, and that's pretty close to T1 limit, which will be difficult or impossible to get over. So the last subject I briefly would like to cover
34:22
is about storing photons. You heard the idea of that in your Davidson's talk yesterday, where he stored them in an atomic gas. We're storing them in a solid. These are presidiumium ions. But the idea is very much the same.
34:40
You store the photons first in an electronic excitation. Now in an excitation, they don't survive very long. So we transfer them to nuclear spin degrees of freedom where they can survive for several milliseconds. The idea is first you store them. You just absorb the input pulse. You apply a photon echo pulse, and you get out the echo.
35:03
And the survival time of this echo is a few microseconds, which is useful, but not great. And this was the experimental scheme for that. And now we transfer the information that was stored in the electronic excitation
35:21
to the nuclear spin degrees of freedom. And then you can also apply dynamical decoupling to the nuclear spins, and you get storage times of a few hundred milliseconds. And this is the experimental data for zero magnetic field in the spin degrees of freedom survival time is a few tens or hundreds of microseconds.
35:45
And we have two different ways to preserve the information in this case. The one I'm showing here is you apply a magnetic field, which is chosen such that it suppresses magnetic fluctuations from the environment.
36:00
I think you call that magic field. In the solid state literature, it's we call the SIVOS condition for zero first order derivative. And then you can also apply dynamical decoupling. I showed that first by comparing the FID with the Hahn echo with the CPMG.
36:21
Obviously, you get the significant improvement in the storage time. And then you can combine the two methods. You apply the magnetic field for at the good condition, and then you apply CPMG. And overall, in this case, we get an improvement in the storage time of about five orders of magnitude,
36:41
which I think is quite nice. So I should come to the conclusion. What I tried to show you was mostly that dynamical decoupling works. I have not discussed the time dependence of the environment and how you can get information on that. If you're interested in that, I can recommend the talk by Gonzalo Alvarez in the same session.
37:04
But here are the conclusions. First of all, dynamical decoupling works. We can extend the coherence lifetime by many orders of magnitude, but you have to be careful. Because if you're doing it the wrong way, then you do more harm than good. But you can compensate for pulse imperfections
37:23
and make the sequences robust. And that's my conclusion. Thank you for your attention. Excellent talk. I wish I had heard that 10 years ago.
37:41
Keep closer. I got a question. The comment is just a more clear clarification in the fence of the way that the QIP community constructed CDD out of asymmetric cycles. I agree there is no overhead at all, and symmetry must be exploited completely. But the other piece of information
38:01
here is that the interaction with the bath is dephasing to begin with. Because if we were to assume depolarization, then the tiniest, there will not be symmetric. They cannot be chosen as symmetric. That's why I said in the beginning, we are only discussing dephasing. The question is in regard to the last part,
38:20
because it was quick. If I remember, there was an experiment on praseodymium on using dynamic decoupling back in 2005. How much beyond how many passes and what more improvement do you get compared to... We're not doing better.
38:40
It's a different material. And this is... Well, I think what they didn't do, they didn't look at the dependence on the initial condition. So I think they did essentially CPMG. Yeah, the phase-alternated CPMG with one state. Yes. I thought this was like in the spirit of Ray's talk
39:02
where we would be repeating- benchmark experiment to see how much better we are doing. No, this is just our first step into that direction. And I mean, this depends a lot on the actual material that you use. And they used a completely different material. Well, it was also praseodymium, but a different host.
39:22
I think I'll do Mike and one more, then we gotta move on. Yes, it helps.
39:43
That was a really nice talk. I had one question about the duty cycle dependence, and I was just curious if you could comment, is that a intrinsic kind of physical phenomenon of some correlation time relative to the duty cycle, or is it something system specific like heating of your RF coil? Well, the initial improvement,
40:00
I mean, if you increase the duty cycle, that depends on the bath correlation time. So, and actually you see different scaling. I mean, if you don't consider pulse imperfection, it always helps to apply more pulses. In other words, increasing the duty cycle. But you get the different scaling behavior depending whether you're below the bath correlation time or above it.
40:22
Does that answer? Okay. Well, but one more question related to that. Eventually you run into finite pulse effects with the duty cycle, and I don't think you considered that much in your analysis, right? Well, you mean in the theoretical or in the experimental analysis? I mean, the duty- Well, obviously it was there in the experiment. I mean, we did the duty cycle almost to one. So we only had small gaps left between the pulses.
40:43
In the analysis that comes in in the sense that your repetition frequency will not go to infinity. And I think he will talk more about that. Okay. Well, let's move on. Let's thank the speaker. Thank you.
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