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Robust Dynamical Decoupling

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and Dortmund will be talking about robust dynamical the coupling yes thank you but I I think you can hear me so it's a pleasure to be here in my 1st Q we conference and I would like to start with introducing the people who did most of the work I'm going to talk about that 1 1 of them is here Connes all actually he's talking later in the same session and the ionic Sunday Martin's this was is not here but he also contributed a lot to their work and I hope I will have time to talk also a little bit about these 2 people or their work work all of which is a graduate student working on optical storage and Jung-hoon calls working environment and the sectors so what I would like to discuss
today is the coherence and the fight against the coherence by dynamical the coupling so what what we have we have hope he'll Cubitt but unfortunates coupled to its environment and that leads to decay of the coherence and of course you would like to do something against that in other words we would like to suppress this coupling and hopefully achieve some longleaf coherence and that's what I would like to discuss in a number of different settings and hopefully convince you that this can actually be done in a
real experiment so I will start with some experimental aspects of dynamical the coupling introduce some robust sequences using friends symmetry in time and then if I still have time to discuss some applications to singles beans and 2 storage of photons in so it so what I'm going to consider here
systems which are subjected to the phasing phase errors now this this can be pictured as by the image you've seen for instance by a Daniels and Daniels talk this be facing and you can really face that by applying the refocusing pulse width it
changes the orientation brand if they process in the same way they will eventually come back to the original location create the Nicole and on the way into the faces that's of course what we want and that's what was 1st achieved by this experiment which also then he'll showed you before the Our of is this this time asymmetric version of the same person I mean do most or most of the experiments
that I will discuss have been done in this sort of lab rats as we like to consider it it's a molecular crystal cold adamant Peynaud molecule is called adamant tain cubits art carbon-13 nuclear spins and they are essentially individual nuclear spins because in each molecule you will mostly at C 0 or 1 of those carbon-13 spins but they see an environment full of protons 1 H nuclear spins and these forests being Bolf they've fluctuating spin boss which actually then causes that the facing the Hamiltonian that we have in this case can be written as an icing Hamiltonian some lower all the protons in the sample and the fact that this is really not a good approximation to pure facing is shown by the ratio of a T 1 longitudinal relaxation time versus dephasing time which is about 10 thousand in this molecule so when you
create coherence and just let it evolve then you see a free induction decay which vanishes on a timescale of a few hundred microseconds obviously that's pretty sure now if you
try to enhance the solution to that you create a holiday code and indeed you can extend the lifetime of the system unfortunately not too long by about a factor of 2 were slightly more than 2 so what's the problem by the signal to leave longer than that and the reason is that these protons in the environment they not static they create the facing which flock to it or be facing rate which fluctuates in time and that's what we have to fight and that the solution against that
is that you don't apply just a single pulse but it career you apply a sequence of pipe pulses and each of those pipe pulses inverts this heteronuclear coupling so you go from plus seniors to plaster minus 2 plus and so on and if you average over all these Hamiltonians calls you get an average Hamiltonian which is 0 in the lowest order the condition for that
to happen if that's the delay between the pulses should be short enough short on the correlation time of the ball in other words in the next pulse has to come before the environment has had time to change significantly and there a lot that can go into this timing off the sequence you've heard about that they might be objects focus and in new Davidson's talk yesterday we hear much more about that in Gonzalez of ours talk it's interesting it can be useful you can adjust the timing for instance by your its formula to improved that the coupling efficiency or you can use it to learn something about the environment as concelebrant show you this is not what I'm going to talk about I will keep all the time in fixed I will have it at identical delays between all the pulses I'm going to talk about
something different but let me 1st state what I want to achieve here what I want to show you our goal is to keep the coherence of the cube eats a life for as long as possible with finite resources that all we have available we're for experimentalists with that we can't have infinite amplitudes and unfortunately our controls on a perfect as you will see so let's see what you can do this is this atom and the molecule the source the free induction decay I showed you before now when a liquor logarithmic scale that was the hot account now let's see what happens if you apply noticing like pulse but the sequence of echo pulses and you see it the rights longer the Tao here is to the later pulses and you see if you make this time shorter the survival of decoherence gets longer so applying more pulses obviously helps that's perhaps not too surprising but let me summarize that this clusters survival time of the free induction decay the harmonic brought an improvement of about a factor of 2 and the more pulses we apply the longer the relaxation or the facing or decoherence time gets this is almost 3 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 limits and in this machine apparently applying more pulses hurts so it seems here the pulses actually destroyed and theorems rather than preserving it and we'll see if you can do something against that but 1st let me discuss another issue that this is related to my question yesterday it in an hour of the new Davidson's talk this measurement was all done with what we call the longitudinal initial conditions that means the initial preparation for instance beings in the direction of the rotation axis of these refocusing pulses and on foot you
can always do that I mean in a pond the computing environment you don't know what the status and so sometimes or in most cases it will not be parallel to the rotation axis so you also have to look at the the the different case what we call transverse initial conditions this is the rotation axis and this is now the orientation of initial magnetization and you see in this case if you apply more pulses it gets worse that coherence time does not increase its decreases and that's because in this regime the pulse imperfections as I said before experiment lists we cannot apply perfect pulses they destroyed a quiz so let's see what we can do and that that's why I was so surprised that the new head homogeneous decay in isotropic decay almost isotropic so here you see the 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 that the simplest cycle of the sea PNG sequence is just 2 pi pulses like and dive market freely chosen the Y. axis as the direction of the rotation so if you have 2 white Yuletide pulses obvious you get the units operate that apart from the overall face now as I said this is not what we do in the lab but what to do in the lab we always apply pipe pulsed plus some additional angle delta which can be positive or negative so this is just the experimental imperfections because the amplifier has noise for instance and do you radio-frequency coral has what's the distribution of the field and so on so if you apply to such pulses obviously don't get the unit operate that any more but you wrote that get now rotation around the axis by an angle to delta why didn't that hurt us in the 1st place only hand if the have the initial condition along the y-axis than the density operator commutes with this overall propagator here and it has no effect on the state but if you 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 be destroyed the coherence and this and the solution to that is very simple actually instead of rotating both times around the same axis you just invert the rotation axis so you combine pipe lost built with minus pipe plus delta at the overall rotation is again 0 as what you what they call this sequence now G 2 and you see here how it works this was the transverse initial condition I showed you before that's the decay 1 of the few dozen pulsars you have no magnetization left but if you alternate the rotation axis you get orders of magnitude improvement to get goin' long-lived Coherence still it's not completely isotropic it's asymmetric because you always rotate around the y-axis sort of I minus Y axis in this case but something that we can learn from that is that it's useful to not to rotate always around the same access because that's a sure recipe that the errors that what you want if you combine different rotation axis and that's exactly what I'm going to talk about now but let me state again the top level we wanted do What's the problem it's in principle if you write down dynamical the coupling you have short pulses and short lend them you can get perfect refocusing now unfortunately we can't do that but as pulses that we have available they have finite strings which means they have finite duration they have flip angle errors as I showed you also you cannot always be exactly resonance for oldest cubits 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 we don't know exactly what they are and that makes it of course hard to compensate the and so what I'm trying to show you as a possible solution if that you use pulses which are insensitive to these experimental imperfections water you combine pulses to robust the sequence is robust dynamical the coupling sequences or if I may have rephrase what you said use imperfect pulses to simulate effects dynamical the couplings I think it was about what you now this was actually
recognize many years ago as this set in a completely different environment mostly looked at the dynamical the coupling refocusing in magnetic resonance imaging and the problem that he had all was very similar he could not make sure that the initial condition was aligned with the rotation axis and the EEG so that if it was aligned yet perfect refocusing but if it was out of phase than the magnetization decay very quickly and the solution which he came up was no instead of using a single rotation axis he just automated the rotation axis between the X. and the Y. axis and the got very good performance independent of the initial conditions and there were several subsequent papers which
improved the statement but let's see how that works this is the performance of the C. P. M. G. sequence in other words each pipe pulse has the same rotation axis for different flip angle errors and we calculate the fidelity after 20 pulses 20 pulses is fairly short sequence but you see with the C. G. sequence you completely lose the signal even if your flip angle error is only 1 % or a few persons so definitely not what you want and if you compare that with the sex life or sequence is see that you can get good performance over a much wider range of experimental imperfections and I'll call I'm going to show you a few other sequences for instance our in our hands the best sequence was this KDD sequence on going to talk about later but let's consider it in a slightly wider context you flip angle errors are under our conditions probably Dement source of error but it's not the only 1 and in many cases you have to consider multiple possible errors and that's what I'm doing here here we have to win the 2 main areas 1 is the flip angle error along the horizontal axis along the vertical axis and have an offset there in other words you cannot apply the pulses exactly at the resonance frequency of 2 cubits and what happens for the CPM cheese sequence is that you very quickly lose the coherence what I've plotted here is color coded the fidelity after pipe pulses and you see again a very small flip angle errors me means you see a signal you can improve that as I said for instance with the ex wife PTT your CD-1 1 sequence and you see these excellent improvement in terms of the fleet battle all it's less impressive in terms of the offset in this theory in the other direction but because it's CDD 1 you can of course at the rate CDD tools and you get again further improvement and actually this is already pretty good if you consider here the highest contour level that means 99 point 9 per cent fidelity after 100 pulses so this is roughly 10 to the minus 5 in our for a single pulse so that looks pretty good but of course we would like to do better and 1 approach for
that it's called composite pulse is a robust pulsars are compensated policies which were introduced in in a MOT by living in Freeman in 1979 and I would say that's pretty similar to what you heard about from the Laurent yesterday was it Monday Monday so so the idea is that instead of using simple pipe pulses the you use robust high pulses and insert them for instance and D 6 sequence replace the simple rectangle apply pulses by composite are compensated high costs now this is the by pulse that are compensated pipe pulse that let it and Freeman introduced this is by the way how it works you want to go from the past that the mind is set and these trajectories here correspond to different flip angle errors and the ideal of peace my pulse in the middle is that it takes you from above the equator to bound to the equator and then you end up close to the south pole no matter how good your RF flip angle it's now this is your most long but there have been improvements in the meantime and for
us in the best pulse to replace this simple pipe turned out to be the sequence of 5 pipe pulses for a number of reasons I'm not going to discuss about it in the literature it sometimes called the fill pulse at the earliest referee found tables in this 1985 paper by rope TECO plans and cook I'm now so you have 5 5 pulses and each of them has a different phase 60 degrees 0 degree 90 degrees euros extending so we take this tide pulse and inserted into the sequences that I showed you before if you use the C. P. G. sequence and just replace each by pulse by 1 of these new pulses then you get an incredible improvement I mean from virtually nothing here you get a pretty large area here where the sequence performs quite well out but you can also insert in in the x for sequence of course and this looks now almost perfect in all these areas here you have the fidelity of 99 comma decimal 9 per cent after 100 titles that's that's nice there is 1 drawback because we replaced each pipe pulse by 5 high pulses means we actually increased the power deposition by a factor of 5 and the question is can we avoid that 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 amenable to be replaced was a pipe pulse and delayed by 5 pulses pulses in the late and obviously that's not the smartest solution but what you can do is use bleak these 5 pipe pulses up and have a 5th of the delay in between so he distributed pulses and delays homogeneously throughout the sequence very wanted a couple now the sequence of 5 pipe pulses replaces the single pipe poles and you can insert the beginning to the life for sequence so you have these 5 pulses 40 x pulse and these 5 pipe pulses for the Y pulse and together this gives you another the coupling cycle actually you have to repeat it twice to get the meals and this be called MKTT KDD that's a sequence I showed before in the one-dimensional bald and just let's take a look at it in the 2 D plot that this
what I was what I showed you before and KDD performs pretty similar to the compensated ex-wife or sequence but the duty cycle here is 5 times lower than here it's the same duty cycle as above here so this is clearly a very nice improvement but so far have only shown new theoretical results these were simulations the only looked at the effect of pulse imperfections there was no environment so in other words we don't have to coupling yet so we have to see if you can actually be coupled with these pulses and that we can see like
that but I show you a preferred to show experimental results here these were the simple CDD sequences CD-1 CD tool and you each day you get improvement many apply more pulses but there's some optimum and then the pulse imperfections takeover now you should replace the pulses in these CD sequences 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 1 duty cycle 1 means you apply only pulses no delays in between duty cycle is to talk duration of the pulses divided by the total duration of the cycle and it's in principle it's important to have good performance for every duty cycle mean high duty cycle is not to be a typical application where you want to store information say quantum memory you don't want to do anything 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 know keeps in all the time and then you have to report lower your duty cycle so in principle it would be good to have something which works so would holes a whole range of duty cycles and that's not yet the case you see here the same pulse sequences performed better than the robust sequences and why why's that so we increase the duty cycle by a factor of 5 here and that's exactly what happened here but if you now take the KDD sequence you see performs will award a whole range and the actually at least for this application it performed best over the whole range of possible applications so and is a turn out to be a quite useful of a way to improve the the coupling performance and get a sequence which turned out to be quite robust under our conditions will discuss a few other ways to improve the coupling performance and always the main focus will be that we don't want to create a lot of poet and of course you can apply harder pulse is putting more power but that that's the Nobel too long and in 1 way that to reduce pulse imperfections orders the simplest way maybe to reduce pulse imperfections is just not apply an impulse so might my teacher PhD supervisor used to say every pulse 1 too many and in a way that's right I mean the best pulses order the least errors are 2 pulses which you don't applied and I would like to show you 1 situation that is is possible because that virtual parts so this is the CDD scheme you start with the X Y for sequence and these are the 1st 2 pulses of the ex-wife Connes and you the insert dx by for sequencing to delayed the mean into the delays between the parts now with virtual pulses I mean that these pulses here we don't really apply we essentially let the pulses act on the pulses that the insert and that means instead of having X Y X Y here we have X minus Y X minus Y and then instead of applying a white pulse here you invert the X pulse and so on and we call that the CCD for
virtual concatenated dynamical decuplet let's see how that works this is the CDD toast sequence again in a two-dimensional representation each point is an experiment very measure the survival probability after 100 pulses as a function of 2 experimental parameters 1 is the offset here and 1 is the spin the pulses and what you see here if you apply your pulses some resonance you get a pretty good performance eventually of course the mechanization decays but for a short pulses spacings and small or a false that it works well but even for small RF offsets use you lose the magnet section and if you now use this virtual pulses instead of the real pulses you get a significant improvement we have a much broader range of offsets over which the magnetizations wise so we have improved the performance in this case with a real action negative overhead this is a comparison with the KDD sequence which were well look similar to the VCE tools in its and of course you can extend that so we have achieved here and improvement of the performance with negative over so we pulled in less power and get her form and no you can't always do that but
sometimes it can be at least you can improve this the performance without putting any overhead that's what I'm going to discuss now if you look at literature on the X or or P. Diddy or 1 sequence you see actually 2 versions of the original 1 looks roughly like this in and this is the 1 which you normally find in the quantum computing literature but it's always the same sequence x y x y with delays how in between the difference is that in the in these old paper that pulsars are placed symmetrically within 1 cycle so you start with the power to delay and end with the power to delay the other version you have a power delay 1st and you end with a pulse of course that doesn't matter if you only look at was stored 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 echo sequence in this version you see twice as many echoes as in the lower from wine while you start with a signal here it decays and 3 faces in the in the window between the pulses in each window you get an here you in the face for a time towel and re- faces after Twitter that's why you get half as many echoes and that already shows you some problem with this basically here the environment has twice as long and twice as much time to change as in the upper sequence and
well this is the experimental comparison that the other was just a plot this is with the symmetric version see the echoes between each pulse and if you apply the ACE immersion you get half as many accounts now the this
is just for a single cell cycle with pool pulses you see that the decay is slightly faster for asymmetric version than for the symmetric it's a small improvement and settling as certainly don't want to say that this is a big improvement but there is 0 were at which seems worthwhile to take
now it also matters if you start to to look for better cycles which are based on that and the idea is that you want to have as a cycle which is time reversal symmetry what that means is if you look at the top and frame Hamiltonian between the pulses and then you compare the sequence of toppling frame Hamiltonians forward in time this the opposite direction than a time reversal symmetry that you get the same sequence and then not the advantage of that if you have such a sequence is that all what the average Hamiltonian terms for a time symmetric secants vanish so again you have something for free there is no overhead associated with that and that's helpful this let's see what that means there
is another I mean that was that is 1 way of saying why you have a slightly longer survival time in the symmetric sequence that I showed you before then in the asymmetric since but there is a important corollary if you now start to combine such cycles and if you calculate the average hamiltonian of such a sequence and you start with the system bus system environment 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 that can be there in experimental imperfections and all these error terms her and that's why
a P the asymmetric sequence has more terms there so it's decays more rapid now every sequence can be improved and a 1 way to improve the sex life or sequence is to inverted in time so you go from x y x y to y x y x and then you put them behind each other so you get the news cycle which is twice as long and which is of course in here time symmetric you can also do that with the asymmetric version and you get now also times symmetric sequence I have ever you started with different sequences and that can also hurt in the combined sequence and you can that for instance you can combine to what these X Y 8 sequences to X Y 16 sequence by combining the makes my 8 Seekins with its hermitian conjugate so how can I have to get a bit faster it seems this is the result of experimental crop and processed tomography you see for the X 5 for sequence the asymmetric and the symmetric perform almost identically but if you combine them to the X 16 sequence here the asymmetric version and here is a smectic person you see that obviously the asymmetric version has an additional error term which causes you precession of the magnetization which destroys the fidelity much faster so but you
can also implement these or use the symmetrized versions for deceiving the sequence in the normal CVD case to insert ex-wife for into the gaps in such a way that pulses here on back-to-back then you can eliminate for instance double pulses if you want and you can also do that for the symmetrized version and you you then insert X Y all sequences in this at 1st so you have delays more equally distributed and at least in our hands that again perform better if used a symmetrized version you see that the symmetric CDD tools sequence performed somewhat better than the asymmetric city tools again it's a small improvement but there's no cost and and that seems to be worse 1 now I have a few minutes left just let me show with few applications of that which are now not on nuclear spins in solid spot on different spins in this case it's says electron spin in diamond you
know these systems when you have 2 free induction decay it vanishes keep time scale of less than a microsecond you can increase that the call to a few tens of
microseconds you can apply PNG sequence to some hundreds of microseconds now in this case we also saw a significant difference for the she sequence depending where the reduced longitudinal or the transverse initial condition hereafter 20 pulses we have very little signal left and if you look at the
function of the angle orientation of initial conditions you see that this clearly gets worse 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 KDD to cater to means through cycles of Haiti which is a 40 pulses and you see we get the pretty independent performance and you can then apply more pulses you increase the analysis and you gets longer T 2 times actually up to a few milliseconds in this case which is quite nice and it stays roughly with the two third power of the number of pulses which you is what you expect if you have a rents spectral widths and I'm sorry I possibly to click the we get a few milliseconds that's pretty close to the 1 limit which really which will be difficult or impossible to get over so the last subject I briefly what like to cover is about storing photons you have at the you hurt the ideals that India Davidson's talk yesterday that he stored them in an atomic gas storing them in a solid these up residue mean irons out but to seek the ideas very much the same you store the photons 1st in India chronic excitation now in in an excitation they don't survive very long so we transferred into nuclear spin degrees of
freedom where they can survive for several milliseconds the idea is 1st you stored and you just absorb the input pulse you apply a photon echo pulse and you read it you get out vehicle and the survival time of the Saco is a few microseconds which is useful but not great and this was the experimental scheme for that and now we transferred the fold the information that was stored in the electronics excitation to the nuclear spin degrees of freedom and then you can also apply dynamical the coupling to the nuclear spins and you get storage times of a few 100 ms and
this is the experimental data 4 0 magnetic field in the spin degrees of freedom the survival time is a few tens or hundreds of microseconds and anything we have 2 different ways to preserve the information in this case the 1 I'm showing here is you applying a magnetic field which is chosen shot such that it suppresses magnetic fluctuations from the environment I think you call that mattered field in the solid state literature is usually called the false condition for sero feel 0 1st or theory and then you can also apply dynamical the coupling I show that 1st by comparing the FIDE with the harmonic cope with a seat Gamgee obviously you get the significant improvement in the storage time and then you can combine the 2 methods you apply the magnetic field 4 at the good condition and then you apply the PNG and overall in this case we get an improvement in the storage time of about 5 orders of magnitude which I think is quite nice so as it should come to this what I tried to show you was
mostly that dynamically coupling birds I have not discussed at the time dependence of the environment and how you can get information that if you're interested in that I can recommend the book by Connes Oliver
as in the same session and but here are the conclusions 1st of all dynamical the coupling birds we can extend the coherence life time by many orders of magnitude but you have to be careful because if you're doing it the wrong way than you do more harm than good but you can compensate for pulse imperfections and make the sequence is robust bands that's my conclusion that you for attention at the excellent OCR
which had heard that 10 years ago you closer and the question of the comment is just the most in verification in defense of the way that the security community constructed CDD out of asymmetric cycles of quietly there is nowhere that all and symmetry must be exploited completely but the the other piece of information here is that there interaction with the biotic defacing to begin with because if we were to assume a depolarization than that they're trying to look at Amendment Nos easy matter that cannot be chosen as well as at the beginning we are on the house and the upper have dared question is in regard to the last part because it was quick hockey fan member there was an experiment on trusted union on using dynamic the coupling bucking 2005 the Solaris yes and I don't know much beyond how many busses and what I morning proven that you get from a to B not doing better it's different material I and this is Nobel died I think what they did do they didn't look at the depends on the initial condition so I think they did a century CPM gene I had a certain it it's in with when status in I think it was like in this he of they an array stock of land it we would be at the keynote handouts 1 experiment to see how much method we this is just the 1st step in that direction and is I mean this depends a lot on the actual material that you use and they used a completely different well it was also in but a different host and of 2 white and 1 more than Denver yes it helps like I was really nice talk I I 1 question about the and the duty cycle dependence I was just curious you could comment is that any intrinsic kind of physical phenomenon of some correlation time relative to the duty cycle or is it something as system-specific liking of your coil well the initial improvement and if you increase the duty cycle that depends on the boss correlation time so and actually use a different scaling if you don't consider pulse imperfection it always helps to apply more the pulses in other words increasing the duty cycle but to get a different scaling behavior depending where the European load above correlation time were above it does that all but 1 more question related to that eventually run the finite pulse effects that duty cycle entity consider that much in your analysis well we need this the core experiment 1 hour and enhances is there an experiment and we we did duty cycle almost to 1 so we have we only had small gaps left between the pulses of a in in the analysis that comes in in the sense that you're you're repetition frequency will not go to infinity and I think he will talk more about that OK well let's move on let's thank the speaker
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Metadaten

Formale Metadaten

Titel Robust Dynamical Decoupling
Serientitel Second International Conference on Quantum Error Correction (QEC11)
Autor Suter, Dieter
Lizenz CC-Namensnennung - keine kommerzielle Nutzung - keine Bearbeitung 3.0 Deutschland:
Sie dürfen das Werk bzw. den Inhalt in unveränderter Form zu jedem legalen und nicht-kommerziellen Zweck nutzen, vervielfältigen, verbreiten und öffentlich zugänglich machen, sofern Sie den Namen des Autors/Rechteinhabers in der von ihm festgelegten Weise nennen.
DOI 10.5446/35325
Herausgeber University of Southern California (USC)
Erscheinungsjahr 2011
Sprache Englisch

Inhaltliche Metadaten

Fachgebiet Informatik, Mathematik, Physik
Abstract Decoherence is among the biggest obstacles for implementing high-performance quantum computing. Possible measures for reducing decoherence include dynamic decoupling (DD), i.e. sequences of inversion pulses applied to the system to be protected. These pulses effectively refocus the interaction between system and environment that drives the coherence decay. For a static environment, they can completely refocus the effect of the environment, resulting in long-lived coherence. We have explored this approach experimentally, using 13C nuclear spins (I=1/2) as the system qubit and a system of coupled 1H nuclear spins as the environment. Our results show that it is possible to extend the coherence time of the system by several orders of magnitude with the help of dynamic decoupling. Care must be taken that the unavoidable imperfections of experimentally realisable rotations do not accumulate throughout the sequence. We therefore designed and tested sequences that compensate the imperfections of the individual pulses over one cycle. The resulting sequences were shown to be very robust and can extend the coherence time of the system by orders of magnitude, independent of the initial state of the system.

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