Dynamical decoupling noise spectroscopy
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Transkript: Englisch(automatisch erzeugt)
00:00
for noise spectroscopy, and other people that is involved in dynamically coupling sequence by using NMR is the people that is mentioned there in the slide. So well, the motivation are more or less common for all of us. What we want to try to do is to try to reduce the level of noise. We want to try to reduce the coherence in order
00:23
to be able to apply quantum error correction codes. So in present technologies, we still need to fight against the coherence to try to achieve the level of noise required for quantum error correction code. One of the ways to attack that is by using decoupling methods
00:41
and try to decoupling the environment from the system. One promising technique is dynamical decoupling. And one of the properties that have dynamical decoupling that we need access only to the system. So we need only to manipulate the system to effectively reduce the interaction with the environment. In order to find the optimal dynamic sequences,
01:00
it's very important to know what is the environment in detail. And this is our motivation. What we want to try to do is to try to measure the characteristic of our environment. So well, I will talk briefly because it was already refreshed today, what is the dynamical decoupling. And for example, this is the pioneer sequence
01:22
that can be identified as a dynamical decoupling sequence and that was the HAN-ECO experiment. And if we have a spin and this spin is interacting with an environment, we look at the survival probability of our initial state as a function of time when this spin is interacting with an environment.
01:40
We'll see that this fidelity starts to decay with time. But if we apply a pipe pulse that invert the orientation of the spin, what we do essentially is to change the system environment interaction, the sign of the system environment interaction, and we will see an echo. So essentially, what we do is to cancel the system environment interaction. And what it keeps is the dynamic of the environment.
02:03
If the environment is static, this echo will be perfect. But if the environment has some dynamic, the echo will not be perfect and the amplitude of the echo will decay. So essentially, in the picture of the runners, what is happening is that all the runners will not achieve the line at the same time.
02:22
So the idea now is to try to decay of this echo will be determined by the spectral properties of the environment. And this is what I want to try to measure. So here, for example, you have the free evolution of our system, how the survival probability decay as a function of time.
02:40
And if we apply a Hanako sequence and we measure the amplitude of the echo, we will get the black squares. And we can see that we can improve the coherence. So in order to characterize the environment, let's assume that we have a purely facing interaction. So the vector of our spin is the SZ operator.
03:02
And let's assume that we are in some kind of semi-classical picture, where the effects of the environment on the spin are some kind of random magnetic field given by this function. So this is some random function that depend on time. And we can characterize this fluctuating field, produced by the environment, by a correlation function.
03:23
And if we produce a Fourier transfer of the correlation function, we can get the spectral density of our environment that will give the different modes that our environment have. So now, let's consider that we have a dynamic decoupling sequence of two pulses.
03:42
And let's consider the sign of the system environment interaction. At the beginning, the sign would be positive. So the sign would be equal to 1. But when we apply a pulse, what we do is to change, effectively, the sign of the system environment interaction. And we have a minus 1 until we apply the second. And we change, again, the sign to 1.
04:01
It was shown some years ago by Kaufman and Kurinsky that there is some kind of filter function disruption. Also, Michael Bierko was talking about that yesterday and near Davidson, that if we look now at the survival probability of our initial state as a function of time, that will decay
04:21
with an exponential function. The argument of that exponential function is given by the overlap of the spectral density of the environment times the Fourier transfer of this sign function of the system environment interaction. So this is what is called the filter function of our dynamic decoupling sequence. And if this filter is very good,
04:42
that will filter some frequency mode of our environment. A little of that was discussed yesterday by Neil Davidson and Michael Bierko. So if we reduce this overlap, we can enhance the decoupling. We can optimize the dynamic decoupling sequence.
05:00
So this is the key, that if we know the spectral density, we can try to find what is a suitable filter function to try to reduce this spectral density decay. So I will not worry about how to optimize that. What I want to try to do is to try to get the spectral density function for that expression. So this is a continuous integral expression.
05:23
And if I want to do this inversion, it's a difficult task. So what we do is to use some kind of tricks. So just to give you an example, let's consider that this green line is the spectral density function. This is some kind of Gaussian spectral density function. I will assume that I am applying two pulses.
05:41
It is a CPMG sequence. And I'm plotting here in a black line, which is the filter function of the CPMG sequence. So you can see we have some kind of oscillation. If we calculate the overlap between the black line and the green line, we will get the decay of our signal after one cycle. But this is a cycle of two pulses. What happens now if we repeat that cycle periodically?
06:03
In the infinite limit, if we repeat a cycle in an infinite number of time, we will get a periodic function. So the Fourier transfer of the sign of the system and volume interaction will be a periodic function. So the Fourier transfer will tend to the Fourier series.
06:21
So for example, here I'm plotting the filter function after 40 cycles. And we are achieving more or less the regime that the Fourier transfer is a Fourier series. And you can see the red line is almost they are sine functions that are centered at the harmonics of the dynamical decoupling sequence. So it's related with the period
06:40
of the dynamical decoupling sequence. So you can see that essentially now the filter function have some magnitude only for discrete values of frequency, that they are related with the harmonic frequency of the dynamical decoupling sequence. So instead of having the continuous expression, the integral that I showed, we will have a discrete sum of spectral densities
07:00
for discrete values of the harmonics of the dynamical decoupling sequence. Another important property of that situation, when we are in this regime that the filter function is a sum of delta functions, what happens is now the decay of the survival probability will be an exponential decay. So we can know that we are already in that regime
07:21
when we start to see that the survival probability is starting to decay exponentially. So now what I can do is to choose different dynamical decoupling sequences, what I can do is to change the delay between pulses in such a way that they start to probe different discrete values
07:41
of frequencies. And I try to choose, if I choose a common multiple for different dynamical decoupling sequences, I can build some kind of linear system of equations. So the idea is here I'm plotting different filter function for different dynamical decoupling sequences, this axis. So all of them have different delay between pulses.
08:00
And what I'm doing is to try to get all the harmonics commensurate between different dynamical decoupling sequence. So all of them have a minimum common multiple. If I do that, I can build this expression. I can write it like a linear system of equation where here I can plot the rate that I get for different dynamical decoupling sequence.
08:21
Here is the spectral density function, and I can build a matrix that is related with the amplitude of the filter function of the harmonics of the dynamical decoupling sequence. Of course, this is assumed that having finite numbers, so we need to make some kind of assumption to try to make that finite. But usually, the spectral density function
08:41
always decay with the frequency. So in some moment, I can neglect the contribution of the tail. Or if I know, if I can determine which is the tail, I can use that assumption to try to put in this expression, and I can invert that. So let's assume that at the beginning that we neglect the contribution of the tail when we can neglect that, and we can make a finite linear system of equation.
09:02
So then we can invert that, and we can find the spectral density function if we measure the rate for different dynamical decoupling sequence. So this is the idea. This is the method. And this is how we can do noise spectroscopy using dynamical decoupling. So now I will give you some examples
09:20
that we perform in NMR setups. So in the inset here, I'm showing you the survival probability of initial state as a function of the number of cycle of CPMG sequences. The different colors are different delay between pulses. So depending on the delay between pulses, we have different decay rates.
09:42
You can see here, this is a log plot. So you can see here that we are already seeing the exponential decay. So we are reading the regime where the model that I showed you works. And if we measure the decay for a different delay between pulses, I can plot here the relaxation time or the decoherent time as a function
10:00
of the pulse delays of differences in the sequences. And you can see where they are the main plot, the square here, the blue square. And you can see, if we start to reduce the pulse delay, we start to see that the decoherent time starts to increase as expected. This region that saturates, this is because of the real pulses that we have in the experiment.
10:24
So let's forget about this part, because we don't have ideal pulses. We cannot approximate our pulses by ideal. But in this region, we can consider that the pulses are more or less ideal. So now, with these relaxations, I can build a linear system of equations.
10:41
And then I can invert it. And I can get the spectral density function. So this is the black square here in the main plot. So here, we have the spectral density as a function of the frequency. In this case, in our system, we can see that the spectral density function has a power law dependence. Essentially, the factor of the power law is almost 4.
11:03
And we can see that the depending of the power law that we have in the tail of the frequency is also the same that we have in the relaxation time as a function of the pulse delay. So already, from here, for a power law spectral density function, we can get the tail of our spectral density
11:24
from the relaxation time. But I want to show you now some more interesting examples. And what we do is now to modify the spectral density of our environment. So the blue lines here are the relaxation time as a function of the delay between pulses.
11:40
This is the previous plot that I showed you before. And then what we do is to apply a radio frequency field to our environment. So our environment is a spin bulb. And when we apply that field to the environment, what we do is to the spin process with a given frequency. This is the frequency that is not there in the lesion. So essentially, what we are doing
12:01
is to put a dominant frequency in our spectral density function. And this is the dominant frequency that the spinner presses. So the different colors are relaxation times as a function of the delays between pulses for different modulating frequencies in our environment.
12:20
So you can see now that the relaxation time changed drastically according to the modulating frequency that we have in our environment. In particular, let's see the black line here. You can see that for this delay between pulses, the coherent time is very short. So what is happening here is that the pulse delay that we are using are putting the system on resonance with the environment.
12:41
So instead of decoupling the environment, what we are doing is to optimize the exchange with the environment. So this is a condition that applying dynamical decoupling with this condition will not be a useful tool. So this is very important. This is showing you very important to know the spectral density of our environment to avoid this kind of situation. So now, if we have all this relaxation time,
13:02
we can build a linear system of equation. We can invert it. And we can get the spectral density function for the different situation. So different colors are the different spectral density functions. You can see that we have the dominant peak at the Rabi frequency of the spin of the environment as expected.
13:21
But something that I want to show you here is if you see, there are some solid symbols and empty symbols. So let's focus on the black line. You can see here that the empty symbols have some kind of maximum. The empty symbols are doing the assumption, as I showed you before, the filter function can have contribution for different harmonics
13:40
of the dynamical decoupling sequence. To obtain the empty symbols, I'm considering that the main harmonic of our filter function is given the decay rate. But the solid is accounting for the addition of all the harmonics of the dynamical decoupling sequence. If we don't consider the contribution for the secondary harmonic, we are observing here
14:01
that we have some kind of maximum. This maximum is not real. That doesn't belong to the real spectral density function. This is because we have the information that is here is not because of the main harmonic. It's because of the secondary harmonics. So if we neglect the contribution of the secondary harmonic, we are getting information of our spectral density that is not real.
14:21
But if we consider the contribution, we can eliminate. Also, I want to cite some recent progress that there's some recent works that they were also published recently. And in particular, for example, the last one was the one that was given by Niv Davieson yesterday. So with all of this, I show you
14:42
how we can use dynamical decoupling for noise spectroscopy. And thank you very much.
15:02
Well, T1, as Dieter show, is the time scale of T1 are very long compared to the time scale of this experiment. And concerning the pulse imperfection was in the experiment that I showed you, we used a CPMG sequence for the Dieter Sutter show also in his talk that CPMG sequence
15:21
is robust against pulse imperfection if the initial state is longitudinal to the pulses. So if we do that, we can remove the pulse imperfection. And essentially, we already did some simulation. And we can see that the behavior of the decay rate is almost compatible with ideal pulses.
15:53
Yeah, sure. The integral of the spectral density is conserved. Sure. Hi, Gonzalo. Yeah. Great talk. Thank you.
16:00
I had a question that's a little out of sync with most of what we've talked about in this conference. Go back to early in your talk, you showed characterization of kind of the natural spectral density of your system. And I think it had 1 over omega to the 3.6 or something. And that's very similar to what
16:21
we saw using a superconducting magnet for a Penning trap experiment, we got 1 over omega to the fourth. And we were curious, I was curious, if you know where this comes from. Is it flux diffusion in the magnet or something like that? I mean, I know that this is not a spin diffusion. Because a spin diffusion, I think you have to,
16:40
I don't remember now exactly, but I think it's a square power law. We don't know exactly what is that. We have some feedback from Edward Feldman. I don't remember. But there are some people that they have some theoretical models that they expect that the tail could be described
17:02
by exponential times a power law. But I don't know exactly where it's coming this factor. We have to explore that. One last question before lunch. In the last case that you have shown of a driven environment, basically to show that if you don't know
17:20
the location of the, well, if you know the spectral density, decoupling can be harmful. Have you tried, that to me seems a very nice setting for trying out those randomized decoupling schemes. Because back in 2005, with Manny and also with my former postdoc, Lea Santos, I was arguing that one instance where randomized scheme
17:41
can be beneficial is they can give you added robustness in the presence of uncertainty in the environment characterization. So on average, you could see good performance. But you mean on average of different spectral densities? No, no, no, no, no, no, the average performance over a randomized decoupling, at each point in time,
18:04
you apply or not apply, and then you have to. If you don't know the optimal time for decoupling, sure. Yeah. We didn't think about it, but it's a good idea. OK, let's thank the speaker and all the speakers of the session. Thank you.