Dynamical decoupling and noise spectroscopy with a superconducting flux qubit

1.5 MB·English

by  Jonas Bylander

Dynamical decoupling and noise spectroscopy
1
1
with a superconducting flux qubit
0
2

n JonasBylander1, SimonGustavsson1,
a
J Fei Yan2,FumikiYoshihara3, KhalilHarrabi3†, GeorgeFitch4,

5 David G. Cory2,5,6, YasunobuNakamura3,7, Jaw-Shen Tsai3,7, WilliamD. Oliver1,4
2

] 1ResearchLaboratoryofElectronics,2DepartmentofNuclearScienceandEngineering,
n
MassachusettsInstituteofTechnology,Cambridge,MA02139,USA.
o
c 3TheInstituteofPhysicalandChemicalResearch(RIKEN),Wako,Saitama351-0198,Japan.
-
r 4MITLincolnLaboratory,244WoodStreet,Lexington,MA02420,USA.
p
5InstituteforQuantumComputingandDept.ofChemistry,UniversityofWaterloo,ON,N2L3G1,Canada.
u
s 6PerimeterInstituteforTheoreticalPhysics,Waterloo,ON,N2J,2W9,Canada.
.
t 7GreenInnovationResearchLaboratories,NECCorporation,Tsukuba,Ibaraki305-8501,Japan.
a
m †Presentaddress:PhysicsDept.,KingFahdUniversityofPetroleum&Minerals,Dhahran31261,SaudiArabia.
-
d January 26,2011
n
o
c
[ Abstract

1
The characterizationand mitigation of decoherencein natural and artificial two-level systems (qubits) is funda-
v
mentaltoquantuminformationscienceanditsapplications. Decoherenceofaquantumsuperpositionstatearises
7
0 fromtheinteractionbetweentheconstituentsystem andtheuncontrolleddegreesoffreedomin itsenvironment.
7 WithinthestandardBloch-Redfieldpictureoftwo-levelsystemdynamics,qubitdecoherenceischaracterizedby
4 two rates: a longitudinal relaxation rate Γ due to the exchange of energy with the environment, and a trans-
1
.
1 verserelaxationrateΓ2 = Γ1/2+Γϕ whichcontainsthepuredephasingrateΓϕ. Irreversibleenergyrelaxation
0 canonlybemitigatedbyreducingtheamountofenvironmentalnoise,reducingthequbit’sinternalsensitivityto
1 thatnoise,orthroughmulti-qubitencodinganderrorcorrectionprotocols(whichalreadypresumeultra-lowerror
1
rates). Incontrast,dephasingisinprinciplereversibleandcanberefocuseddynamicallythroughtheapplication
:
v ofcoherentcontrolpulsemethods[1–3]. Inthisworkwedemonstratehowdynamical-decouplingtechniquescan
i
X moderatethe dephasingeffectsof low-frequencynoise on a superconductingqubit[4–6] with energy-relaxation
time T = 1/Γ = 12µs. Using the CPMG sequence[7,8] with upto 200π-pulses, we demonstratea 50-fold
r 1 1
a improvement in the transverse relaxation time T over its baseline value. We observe relaxation-limited times
2
TCPMG = 23µs 2T resulting from CPMG-mediated Gaussian pure-dephasing times in apparent excess
2 ≈ 1
of 100µs. We leverage the filtering property of this sequence in conjunction with Rabi and energy relaxation
measurementstofacilitatethespectroscopyandreconstructionoftheenvironmentalnoisepowerspectraldensity
(PSD)[9,10].
1
Severalmulti-pulsesequencesdevelopedwithinthe field ofnuclearmagneticresonance[2] (NMR) have re-
cently been applied to mitigate noise in qubits based on atomic ensembles [11], semiconductor quantum dots
[12,13],anddiamondnitrogen–vacancycentres[14,15].Weextendthesemethodstotherealmofsuperconducting
quantumdevices,andsubjectaremarkablylong-livedqubittovaryinglevelsoflongitudinalandtransversenoise
byrotatingthequbit’squantizationaxis, againstwhichwe characterizethebaseline coherenceratesΓ , Γ , and
1 2
Γ . We evaluatethreedynamical-decouplingpulseprotocols: theCarr-Purcell[7](CP);Carr-Purcell-Meiboom-
ϕ
Gill[8](CPMG);andUhrigdynamical-decoupling[16](UDD)sequences. Thenarrow-bandfilteringpropertyof
theCPMGsequenceenablesustosampleenvironmentalnoiseoverabroadfrequencyrange0.2–20MHz,andwe
observea1/fα-typespectrumwhichweindependentlyconfirmwithaRabi-spectroscopyapproach. Wefurther-
morecharacterizetheenvironmentalnoisefrom5.4to21GHzbymonitoringthequbit’srelaxationrate[9,10].
Thedeviceisapersistent-currentqubit(Figs.1aandA1),analuminiumloopinterruptedbyfourAl-AlO -Al
x
Josephsonjunctions. WhenanexternalmagneticfluxΦthreadingtheloopisclosetohalfasuperconductingflux
quantumΦ /2, thediabaticstates correspondtoclockwise(counterclockwise)persistentcurrentsI = 0.18µA
0 p
withenergies ¯hε/2 = I Φ ,tunablebythefluxbiasΦ = Φ Φ /2. AtΦ = 0,thedegeneratepersistent-
p b b 0 b
± ± −
currentstates hybridizewith a strength ¯h∆ = h 5.3662GHz (Figs. 1b and A2b), where ¯h = h/2π and h is
×
Planck’sconstant. Thecorrespondingtwo-levelHamiltonianis[4,5]
¯h
ˆ = [(ε+δε)σˆ +(∆+δ∆)σˆ ], (1)
x z
H −2
whichincludesnoisefluctuationtermsδε andδ∆, andσˆ arePaulioperators(Fig.A2). Theground(0 )and
x,z
| i
excited (1 ) states have frequency splitting ω = √ε2+∆2 and are well isolated owing to the qubit’s large
01
| i
anharmonicity,ω /ω 5. Theenvironmentalnoiseleadingtofluctuationsδε (e.g.,fluxnoise)andδ∆(e.g.,
12 01
≈
criticalcurrentandchargenoise)physicallycouplestothequbitintheε–∆frame(equation1). However,their
manifestationaslongitudinalnoise(dephasing)ortransversenoise(energyrelaxation)istunable[17]bytheflux
biasΦbanddetermined,respectively,bytheirprojectionsδωz′ ontothequbit’squantizationaxisσˆz′ (whichmakes
anangleθ =arctan(ε/∆)withσˆz)andδω⊥′ ontotheplaneperpendiculartoσˆz′.
Thechipismountedina3He/4Hedilutionrefrigeratorwith12-mKbasetemperature. Foreachexperimental
trial, we initialize the qubit by waiting sufficient time ( 1 ms) for it to relax to its ground state. We drive the
∼
desiredquantum-staterotationsof angleΘ by applyingcalibratedin-phase(X ) and quadrature(Y ) harmonic
Θ Θ
fluxpulsestothequbitloop.ThepulsescompriseGaussianenvelopeswithatypicalstandarddeviationσ =1.2ns
and truncated at 3σ. The qubit readout has 79% visibility (Fig. A3) and is performed in the energy basis by
±
determiningtheswitchingprobabilityP ofa hystereticdc SQUID, averagingoverseveralthousandtrials(see
sw
Appendix).
We begin with a spectroscopic characterization of our device and its baseline coherence times. The qubit
level splitting ω is measured via saturated frequency spectroscopy (Fig. 1b), and at low-power it exhibits a
01
Lorentzianfull-width-at-half-maximum(FWHM) linewidth ∆f = 0.18MHz at Φ = 0 (Fig. 1c). The
(FWHM) b
energyrelaxationisgenerallyexponentialanditstimeconstantT =12 1µs(Fig.1d)isremarkablylongamong
1
±
superconductingqubits[6],afeatureweleverageinthisworkinconjunctionwithquantizationaxistunability.We
observe similarly long decay times at Φ = 0 for the Hahn spin-echo, T = 23µs (Fig. 1d), Ramsey free
b 2,E
∗
induction,T = 2.5µs(Fig.1e), andRabioscillations,T = 13µs(Fig.1f). Althoughthe spinechoandRabi
2 R
exhibitanapparentlyexponentialdecayatΦ =0(theyareessentiallyT -limitedatthisfluxbias),ingeneral,their
b 1
decayfunctionsarenon-exponential.Furthermore,thedephasingtimesdecrease(ratesincrease)dramaticallyaway
fromΦ =0(Figs.2cand3b)duetothequbit’sincreasedsensitivitytothedominantδε-noise(fluxnoise)inthis
b
system[18](increased ∂ω /∂ε),which,aswewilldemonstrate,canbemitigatedwithmulti-pulsedynamical-
01
| |
decouplingsequencestopushcoherencetimesuptowardstheT -limit.
1
Decoherencein superconductingqubitshasbeen studied theoretically[9,17,19,20] and experimentally[10,
17,18]. EachnoisesourceλischaracterizedbyitsPSD,whichquantifiesthefrequencydistributionofthenoise
∞
power, S (ω) = (1/2π) dt λ(0)λ(t) exp( iωt). In contrast to a simple Bloch-Redfield picture, which
λ −∞ h i −
presumesweaklycouplednoisesourceswithshortcorrelationtimesτ T ,T resultinginpurelyexponential
c 1 2
R ≪
2
a b c d

R R L C Hz)5.9 1Psw0.99 1 Relaxation: Xπ τ read outt
V I G 0.98 P
ΦdVc + Imdwc + pulse S|Q0U/1ID\ fFrequency, (55..75 0.50.97 Df ( F W H M ) = 0.18 MHz s0w.5 Echo: Xπ/τ2/2 XπTT2 τ1 E / 2 == 12X23 π mm/s2st
5.3 0 0.96 0
-2 -1 0 1 2 5.365 5.367 0 20 40 60
Flux detuning, Φ (mΦ ) Frequency, f (GHz) τ (ms)
e f b 0
P 1 Ramsey: Xπ/2τ Xπ/2t Rabi oscillations TR = 13 ms |0\
sw
0.5
X
Θ(τ)
T2 * = 2.5 ms t |1\
0
0 2 4 6 0 5 10 15 20 25
τ (ms) τ (ms)
Figure1:Qubitdeviceandcharacterization.a,Deviceandbiasingschematic:Analuminiumsuper-
conductingloopinterruptedbyJosephsonjunctions(crosses)witharead-outdcSQUID.b,Frequency
spectroscopyof the qubit’s 0 1 transition. c, Spectroscopyat Φ = 0 (arrowin b). d, Echo
b
| i → | i
decay(bluetriangles)andrelaxationfromtheexcitedstate(blackdots)atΦ =0.Intheinsets,τ isa
b
timedelayandX symbolizesarotationoftheBlochvectorbytheangleΘaroundtheaxisσˆ . The
Θ x
redsquaresindicatethetimeofreadout. e,Free-inductiondecay(Ramseyfringe)atΦ =0. f,Rabi
b
oscillationsatΦ =0.
b
decayfunctions, moregenerallythe decayfunctionsare non-exponentialfornoise sourceswith longcorrelation
times,singularnearω 0(e.g.,1/f-typenoiseatlowfrequencies,relevanttodephasing).
≈
Asuperpositionstate’saccumulatedphaseϕ(t)= ω t+δϕ(t)diffusesduetoadiabaticfluctuationsofthe
01
transitionfrequency,δϕ(t) = (∂ω /∂λ) tdt′δλ(t′).hForinoisegeneratedbyalargenumberoffluctuatorsthat
01 0
areweaklycoupledtothequbit,itsstatisticsareGaussian.Ensembleaveragingoverallrealizationsofthestochas-
R
tic process δλ(t), and taking the sources λ to be independent, the dephasing is exp[iδϕ(t)] exp[ χ (t)],
N
h i ≡ −
withthecoherenceintegral
∂ω 2 ∞
χ (τ)=τ2 01 dωS (ω)g (ω,τ), (2)
N λ N
∂λ
λ (cid:18) (cid:19) Z0
X
where τ is the free evolution time and N will denote the number of π pulses in the pulse sequences [11,16].
Equation (2) expressesthat the PSD S (ω) is filtered by a dimensionless weightingfunction g determinedby
λ N
thepulsesequence,andtheaggregatedλ-noisetranslatestodephasingthrough∂ω /∂λ,thequbit’slongitudinal
01
sensitivity to λ-noise. For Gaussian noise with spectral distribution S (ω) = A /ω at low frequencies, the
λ λ
coherenceintegralresultsinaGaussiandecayfunction,χ (τ)=(Γ τ)2.
N ϕ
The Ramsey free-inductionand Hahn spin-echo dependenceon flux bias (Fig. 2c) are both apparently con-
sistent with Gaussian distributed, 1/f-type noise. Ramsey free induction, the free evolution of a superposition
state for a time τ (Fig. 2a with no π pulses), has a filter functiong peakedat ω = 0 (Fig. 2b) and is sensitive
0
to low-frequency longitudinalnoise δωz′(ω 0). Inhomogeneitiesin the precession frequency ω01 from one
→
realizationofthepulsesequenceto thenextleadtoa decayoftheaveragedsignal. We denotesuchfluctuations
“quasi-static” noise and characterizethem by a noise variance σ2 = 2 ωuλv dωS (ω), with cut-offfrequencies
λ ωiλr λ
ωiλrandωuλvdeterminedrespectivelybytheaveragingtimeoveralltrialsRandthetypicalfree-evolutiontimeduring
3
asingletrial. Incontrast,theHahnspin-echosequence[1],asingleπpulseappliedattimeτ/2(Fig.2awithoneπ
pulse),hasafilterfunctiong peakedawayfromω =0(Fig.2b)andislesssensitivetoquasi-staticnoise.Weplot
1
thedecayrates1/T versusfluxbiasΦ forRamseyandechoinFig.2c(seealsoFig.A4),where,forpurposesof
e b
comparisonamongstdifferentdecayenvelopes,T parameterizesthetimeT todecaybyafactor1/eindependent
e 2
oftheexactdecayfunction(seeAppendix). AtΦ = 0, δ∆-noiseisthe dominantlongitudinalnoisethatlimits
b
theRamseydecay,yetitisrefocusedwithasingleπ-pulseresultingintheT -limitedexponentialecho-decayin
1
Fig.1d. As Φ isincreased,boththeRamseyandechodecayratesincreaseduetothequbit’sincreasedlongi-
b
| |
tudinalsensitivitytoδε-noise(seeFigs.2candA4). Theδε-noiseistoolargefortheechotorefocusefficiently,
duetoitshigh-frequencytail,andwefindbest-fitphasedecayfunctionsthatareGaussian,χ(τ) = (Γ τ)2.
ϕ,F(E)
We extract the ratio Γ (Φ )/Γ (Φ ) 4.5, as expected for 1/f noise [17], with an equivalent flux-noise
ϕ,F b ϕ,E b
≈
amplitude[18]A = (1.7µΦ )2. Importantly,wenotethatinthisanalysisandrelatedworks[17,18], thePSD
Φ 0
waspresumedapriori totaketheform1/fαwithα=1.
Numerical simulations (see Appendix), including the measured T decay at each Φ , linearly coupled and
1 b
uncorrelatedquasi-static noises, and uncorrelateddynamic1/f noise from104 to 1010 Hz, reproducethe entire
∗
Φ -dependenceofT andT usingtheparametersinTable1andarealsoconsistentwithequation(2).
b 2 2,E
Inorderto mitigate higher-frequencynoise thanwhatthe Hahnspin-echocan efficientlyrefocus, we further
shape the filter function by applyingadditionalπ-pulses (Figs. 2a and A5). The filter g (ω,τ) dependson the
N
numberN anddistributionofπ-pulses[11,16,21,22]duringthetotalsequencelengthτ,
N
1 2
g (ω,τ)= 1+( 1)1+Nexp(iωτ)+2 ( 1)jexp(iωδ τ)cos(ωτ /2) , (3)
N (ωτ)2 − − j π
(cid:12) Xj=1 (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
whereδ [0,1]isthenormalizedpositionofthecentreofthejthπ-pulsebetweenthetwoπ/2-pulses,andτ is
j π
∈
thelengthofeachπ-pulse.Asthenumberofpulsesincreasesforfixedτ,thefilterfunction’speakshiftstohigher
frequencies(Fig. 2b), leadingto a reductionin the netintegratednoise(equation2) for1/fα-typenoise spectra
withα>0. Alternatively,forafixedtimeseparationτ′ =τ/N (validforN 1),thefilterasymptoticallypeaks
≥
nearω′ =2π/4τ′asmorepulsesareadded.Inprinciple,onecanadaptthefilterfunctiontosuitaparticularnoise
spectrumviathechoiceofdynamical-decouplingprotocol.
Wehaveevaluatedthreedifferentdynamical-decouplingprotocolsrelevantfor1/f-typepowerlawnoisespec-
tra[23–25]. TheCPandCPMGsequences[7,8]aremulti-pulseextensionsoftheHahnechowithequallyspaced
π-pulseswhosephasesdifferfromtheinitialπ/2pulseby0◦(X )and90◦(Y ),respectively(Fig.2a).TheUDD
π π
sequence[16]hasY -pulsepositionsdefinedbyδ =sin2 πj .
π j 2N+2
InFig.2c,weincludethe1/TedecayratesforCPMGd(cid:16)ynamic(cid:17)al-decouplingsequenceswithN =2...48π-
pulsesalongwiththeRamsey(N =0)andHahnecho(N =1)alreadydiscussed.Thedecayratesmonotonically
improve towards the 1/2T -limit as the number of π-pulses increases, extending the range around Φ = 0 for
1 b
whichdephasingisnegligible.Outsidethisrange,asthequbit’ssensitivitytoδε(flux)noisegrowswithincreasing
Φ ,increasingthenumberofπ-pulsesmitigatesthenoisesufficientlywelltoachieveadesireddecayrate.
b
| |
At a specific flux bias Φ = 0.4 mΦ , where the qubitis highly sensitive to δε noise, the CPMG sequence
b 0
givesamarkedimprovementinthedecaytimeT uptoN 200π-pulses(Fig.2d),beyondwhichpulseerrors
e
begintolimittheCPMGefficiency.Weachievea50-folden≈hancementofTCPMGovertheRamseyT∗ (Fig.2d),
2 2
and well over 100-fold improvement in the Gaussian pure dephasing time T . The CPMG sequence performs
ϕ
about5%betterthanUDD,indicatingthatthe1/f δε-noisespectrumexhibitsarelativelysoft(ifany)ultraviolet
cutoff [21,25], and it dramatically outperformsCP, as Y -pulse errorsappear only to fourth order with CPMG,
π
whereaswithCP,X errorsaccumulatetosecondorder[26].
π
WeusethefilteringpropertyoftheCPMGsequencetocharacterizetheδε-noisespectrum.Thefilterg (ω,τ)
N
is sufficiently narrow about ω′ that we can treat the noise as constant within its bandwidth B and approximate
equation (2) as χ (τ) τ2(∂ω /∂Φ)2S (ω′)g (ω′,τ′)B (Fig. A6). We compute ω′ and B numerically
N 01 ε N
≈
for each N and τ used in the CPMG measurements of Fig. 2d. The measured decay function contains three
decay rates: dephasing Γ(N) and exponentialrelaxation Γ /2 during the total free-evolutiontime τ, and pulse-
ϕ 1
4
induced decay Γ during Nτ (we assume Γ to be independentof N). Conceptually, Figs. 2e–g illustrate for
p π p
N = 84howwedivideouttheΓ andΓ (assumingexponentialpulse-induceddecay)componentsfromtheraw
1 p
data (e)to obtainthe Gaussianphase decay(f), andthencomputeS (ω)(g). Morerigorously,we onlyuse that
ε
methodtodetermineastartingpointatasinglefrequency,andthenusearecursivemethodtoobtaintheremainder
of the spectrum without presuming a functional form for the pulse-induced decay, the dephasing, or the noise
a Xπ/2 Yπ Yπ Yπ Yπ Yπ Yπ Xπ/2 b 1 N = 0 τ = 1 ms
2τN Nτ Nτ . . . τN τN 2 τN ω,τ) 12 4 . . . 10 . . .
N π-pulses (N >_ 1) t g (N
0
c ε/2π (MHz) 0 2.5 5
-400 -200 0 200 400 Frequency, f (MHz)
1
Ramsey CPMG e 0.8
0.8 N increases Φ b = -0.4 mΦ0
w
Ps N = 84
-1s ) 0.6 1 - 0.6
m
T (e 0.4
/
1
0.4
f
0.2 1
\
0 1/2T1 tϕ() ]
-0.4 -0F.l2u x d e t u n i n 0g , Φ ( m Φ 0 .2) 0.4 ip[ 0.5
b 0 x
d e
\
2T1 0
0 10 20
s)10 CPMG Free evolution, τ (ms)
m
T (e simulation g 8
me, CPMG 10
e1/ decay ti 1 Ramsey (N=0) CUPDD -1S ω() (s )ε110067
Φ = -0.4 mΦ
b 5
0.1 10 6 7 8
1 10 100 1000 10 10 10
Number of π pulses, N Frequency, f (Hz)
Figure 2: Dynamical decoupling pulse sequences. a–b, CPMG pulse sequence and filter
function g (ω,τ) for a fixed total pulse-sequence length τ. c, Decay rates (inverse of 1/e
N
times) vs. flux detuning: Free induction (green squares) and CPMG (coloured dots) with N =
1,2,4,6,8,10,16,20,24,30,36,42,48. Solid lines are calculationsusing equation(2) with param-
eters in Table 1. d–g, Measurements at Φ = 0.4mΦ (ε/2π = 430MHz). d, 1/e decay time
b 0
−
under N-pulse CPMG, CP, and UDD sequences. The simulation (red line) assumes perfect pulses.
e, Population decay under an 84-pulse CPMG sequence. Gaussian fit with fixed exponentialdecay
contributionsT =12µs,T =1.75µs.f,PhasedecayofthesignalineafterdividingoutT andT .
1 p 1 p
g,NoisePSDcalculatedfromthedatainf.
5
spectrum(seeAppendix). Bothapproachesyieldnearlyidentical1/fα-typespectrawithaslightincreaseinthe
measuredPSDabove2MHz;weplottherecursivelyextractedPSDS (ω)overtheregion0.2–20MHzinFig.4.
ε
Interestingly,thePSDestimatedinthismannerisbetterapproximatedbya1/fαpowerlaw[27]withα=0.9<1
(solid,redline)withnoiseamplitudeA = (0.8µΦ )2,obtainedbyfittingthelower-frequency,linearportionof
Φ 0
thePSD.Projectingthislinetohigherfrequenciescomeswithinafactor2ofthetransversenoiseatfrequency∆
asextractedfromtherelaxationmeasurementsdescribedbelow.
Weconfirmedthespectrumoverasimilarfrequencyrangebyanalyzingthedecoherenceduringdrivenevolu-
tion,whichprovidesanindependentmeanstorotatethequantizationaxiswithrespecttothenoisesourceswhen
viewed in the rotatingframe [17,28]. A transversedrivingfield at frequencyω results in Rabi oscillationswith
angularfrequencyΩ = Ω2+(∆ω)2 Ω+(∆ω)2/2Ω,where∆ω = ω ω . Integratingtheoscillations
R 01
≈ −
overa normaldistributionwithvarianceσ2, weobtainthequasi-staticdecayfunctionζ(τ) = 1+(uτ)2 −1/4,
p ε
whereu=(ε/ω )2σ2/Ω. AlongwithΓ ,thenoiseattheRabifrequencyΓ(λ) =πS (Ω )comprisestheusual
01 ε 1 Ω λ R (cid:0) (cid:1)
exponentialRabi-decayrate
3 1 ε 2 1
Γ = Γ + Γ(∆) cos2θ+ Γ(ε), (4)
R 4 1 2 Ω ω 2 Ω
(cid:18) (cid:19) (cid:18) 01(cid:19)
wherecos2θ 1asthequantizationangleissmall. Thecombineddecayfunctionisζ(τ)exp( Γ τ) (Fig.3a;
R
≈ −
seealsoAppendix).
To determine S (Ω ), we measured the Rabi oscillations vs. Φ with fixed Rabi frequency Ω . For each
λ R b R
Ω ,wefindtheε-independentpartoftheratebyfittingtheenvelopeoftheoscillationsatε = 0. TherateΓ(∆)
R Ω
a
1
w
Ps
0.6
0.2
0 2 4
Time, τ (ms)
b
0.2
-1s )0.1
m
(ΓR
0
-0.5 0 0.5
Detuning, Φ (mΦ )
b 0
Figure 3: Decoherence during driven dynamics. a, Rabi oscillations with Ω /2π = 2MHz at
R
ε/2π = 225MHz. Theredlineenvelopeisafittingusingζ(t)andΓ . TheblacklineshowstheΓ
R R
decayonly,andthegreenlinetheζ(t)-envelopecontribution.b,Rabi-decayrateΓ vs. fluxdetuning
R
atΩ /2π=2MHz,withaparabolicfittoequation(4)usedtoobtainS (Ω )= 1 Γ(ε).
R ε R π Ω
6
was too small to distinguishaccuratelyfromΓ , consistentwith its correspondinglysmall quasi-static noise σ2
1 ∆
(Table1). Then,forε = 0,wedivideouttheknownquasi-staticcontributionζ(τ)andfittotheparabolictermin
6
equation(4),fromwhichweobtainΓ(ε) (Fig.3b). Usingthisapproach,wefindS (Ω)tobeconsistentwiththe
Ω ε
1/fαnoiseobtainedfromtheCPMGmeasurements(Fig.4).
We now turn to transversenoise, i.e. δε-noiseat ε = 0, andδ∆-noise at ε ∆ (Fig. 4, inset), at the qubit
≫
frequency ω responsible for energy relaxation Γ . In the low-temperature limit, k T ¯hω , where the
01 1 B 01
≪
environmentcannot excite the qubit, the golden-rule expression for Γ in a weakly damped quantum two-level
1
systemis
2
Γ = π ∂ω⊥′ S (ω )= πS(ω ), (5)
1 λ 01 01
2 ∂λ 2
λ∈δε,δ∆(cid:18) (cid:19)
X
with∂ω⊥′/∂λthequbit’ssensitivitytotransversenoiseandS(ω01)thetotalPSD(seeAppendix).
Weapplyalong( T )microwavepulsetosaturatethetransitionandmonitortheenergydecaytotheground
1
≫
state,usingequation(5)todetermineS(ω )overthefrequencyrange∆ ω 2π 21GHzbytuningΦ . At
01 01 b
≤ ≤ ×
Φ =0(ω =∆)andusingameasurement-repetitionperiodt >1ms,weobserveT =12 1µsasshown
b 01 rep 1
±
inFig1d.Ast becomesshorterthan1ms,thedecaybecomesincreasinglynon-exponential,whichweattribute
rep
totheresidualpresenceofnon-equilibriumquasiparticlesgeneratedbytheswitchingSQUIDduringreadout. We
observe structure in the Γ data due to environmental modes (e.g. cavity modes, impedance resonances) with
1
uncontrolledcouplingstothequbit(Fig.4). Forcomparison,weplottheexpectedJohnson-Nyquistfluxnoisein
Fig.4duetotheR=1/G=50ΩenvironmentmutuallycoupledwithstrengthM =0.02pHtothequbitviathe
microwaveline,
2
1 ∂ε 2h¯ωG
SJN(ω)= M2 . (6)
ε 2π ∂Φ 1 e−h¯ω/kBT
(cid:18) (cid:19) −
Thisknownnoisesourcefallsabout100timesbelowthemeasuredPSDatf = ∆/2π(reddotinFig.4)where
01
the relaxation is due solely to δε noise, indicatingthat the dominantsource of energyrelaxationlies elsewhere.
The crossoverf between the effective 1/f- and f-type flux noises (Fig. 4) occurs between ∆/2π and k T/h,
c B
whereT =50mKistheapproximateelectronictemperatureofourdevice[10,30].
Table1:Quasi-staticnoiseparametersusedinsimulations,andcoherencetimes.
Noiseparameters σ /2π ωλ/2π ωλ /2π A
λ ir uv λ
λ=ε (equiv. Φnoise) 10MHz 1Hz 1MHz (1.7 10−6)2 Φ2
× 0
λ=∆ (equiv. I/I noise) 0.06MHz 1Hz <0.1MHz (4.0 10−6)2
c
×
Coherencetimes T T∗ TCPMG TCPMG/T∗
1 2 2 2 2
Φ =0mΦ 12µs 2.5µs 23µs(N =1) 9
b 0
Φ =0.4mΦ 12µs 0.27µs 13µs(N =200) 48
b 0
In Ramsey-fringeand Hahn-echosimulations, we describe the Gaussian noise distributionsby their
standarddeviations,σ , obtainedbyintegratingthe 1/f noisesoverthebandwidthgivenbytheex-
λ
perimentalprotocol(cut-offfrequenciesωλandωε ,seetext).Atε=0,thedephasingimprovement
ir uv
under a Hahn echo is greater than the theory would suggest for δ∆ 1/f noise that extends to high
frequencies; the lower ω∆ gives consistency (see Appendix). The equivalent flux and normalized
uv
critical-currentnoiseamplitudes,A ,arevaluesderivedfromtheRamseyandechodataassuminga
λ
powerlaw 1/fα with α = 1 and thatall noisein ε and∆ is flux andcritical-currentnoise, respec-
tively;theyareconsistentwithpreviouslyreportedvalues[18,29]. Usingtheseparametersinsimula-
tionsyieldedagreementwithN-pulsedynamical-decouplingdata,consistentwithequation(2). The
coherencetimesaregivenattwobiaspointsdominatedbyδ∆andδεnoise,respectively.
7
Thedynamical-decouplingprotocolsdemonstratedinthisworkcomprisethesametypesofsimplepulsesthat
are used for quantum gate operations and therefore require little additional overhead to implement. Integrating
refocusing pulses into qubit control sequences, e.g., by forming composite gates that incorporateboth quantum
operationsandrefocusingpulses,willleadtolowerneterrorratesinsystemslimitedbydephasing[31]. Despite
observinglevelsof1/f fluxandcritical-currentnoisesimilar tothoseobservedubiquitouslyinsuperconducting
qubitsandSQUIDs[6,29],wecouldmitigatethisnoisedynamicallytoincreasethepuredephasingtimesbeyond
8
10
T relaxation
1
7
10
noise in ε:
Rabi spectroscopy ( )
106 CPMG spectroscopy (dots) Effective 1/f α flux noise
-1s ) α = 0.9
ω) (105
SD, ( 1/f Transverse noise α = 1 Effective
PS104 Z = Z’ Z noise
Noise noise in D 1/(fββ>_2) Y = Y’ θ Y = YZ’’
3
10 X = X’ X
2 ε = 0 ε = 4D X’
10
Expected Johnson-Nyquist flux noise ~ k T/h D/2π
B
1
10
5 6 7 8 9 10
10 10 10 10 10 10
Frequency, f (Hz)
Figure4: Noise-power spectraldensity (PSD). Multi-coloureddots, δε-noisePSD (0.2–20MHz)
derived from CPMG data at Φ = 0.4mΦ (see text). Colours correspond to the various N in
b 0
−
Fig. 2b–c; grey dots for data up to N = 250. Yellow squares, δε-noise PSD (2–20MHz) derived
fromRabispectroscopy(Fig.3;seetext). Diagonal,dashedlines,Estimated1/f flux(red)andδ∆
(blue)noiseinferredfromtheRamseyandechomeasurements(cf.A andω∆ inTable1).Solid,red
λ uv
line,Power-lawdependence,S (f)=A′/(2πf)α,extrapolatedbeyondthequbit’sfrequency,∆/2π.
ε ε
TheparametersA′ = (0.8µΦ )2 andα = 0.9weredeterminedbyfittingthelow-frequency,linear
ε 0
portionoftheCPMGPSDdatabeforetheslightupturnbeyond2MHz(seeAppendix). Theshaded
area covers α 0.05. Green dots, High-frequencyε and ∆ PSD inferred from energy-relaxation
±
measurementsabove ∆/2π=5.4 GHz. Purple line, Guide to indicate linearly increasing Nyquist
(quantum)noise, including the eigenbasis rotation (see inset); dots indicate transverse δε (red) and
δ∆(blue)noises. Inset,Graphicrepresentationofthequantizationaxis(greyarrowsoffixedlength)
with the qubit’s(Z′) eigenstate tilted fromthe “laboratory”frame(Z) bythe angle θ. Fieldsε(Φ )
b
and∆pointintheX andZ directions,respectively. Redandbluedouble-arrowsindicatetransverse
noise.
8
0.1ms,morethanafactor104 longerthantheintrinsicpulselength. However,despitehavingaremarkablylong
energyrelaxationtimeT = 12 µs, thetransverserelaxationT 2T wasultimatelylimitedbyit. Dynamical
1 2 1
≈
decouplingprotocolsgoalongwaytorefocusingexistinglevelsof1/f noiseandachievelongcoherencetimes,
andthemainemphasisisnowonidentifyingandmitigatingthenoisesource(s)thatcauseenergyrelaxation. We
noteforfurtherstudythatthePSDpowerlawobtainedexperimentallybytheCPMGtechnique,whenextendedto
higherfrequencies,fallswithin a factortwo ofthe measuredtransversenoise, suggestingthe possibilitythatthe
microscopic mechanism responsible for low-frequencydephasing may also play a role in high-frequencyrelax-
ation.
Appendix
Measurement set-up.
WeperformedourexperimentsatMIT,inadilutionrefrigeratorwithabasetemperatureof12mK.Thedevicewas
magneticallyshieldedwith 4 Cryoperm-10cylindersand a superconductingenclosure. Allelectricalleadswere
attenuatedand/orfilteredtominimizenoise.
WeusedtheAgilentE8267Dmicrowavesource,andemployedtheTektronixAWG5014arbitrarywaveform
generatortocreateI/Qmodulatedpulses,andtoshapetheread-outpulse.
Description ofthequbit.
WefabricatedourdeviceatNEC,usingthestandardDolanangle-evaporationdepositionprocessofAl–AlO –Al
x
onaSiO /Siwafer(Fig.A1a).
2
Thepersistent-current,orfluxqubit[4,5,32,33]consistsofasuperconductingloopwithdiameterd 2µm,
∼
interruptedbyfourJosephsonjunctions(Fig.A1). ThreeofthejunctionseachhavetheJosephsonenergyE =
J
210GHz,andchargingenergyE = 4GHz;theforthissmallerbyafactorα =0.54. Theratioofenergyscales
C
putsthedeviceinthefluxlimit,E /E 50,thusmakingthephasesacrosstheJosephsonjunctionswelldefined.
J C
≈
ThegeometricandkineticloopinductancesarenegligiblecomparedtotheJosephsoninductance:
L µ d 2pH, L =µ λ2 l/S 30pH, L =Φ /2πI 10nH,
g ∼ 0 ∼ k 0 L ∼ J 0 c ∼
whereweusedλ =100nm,l=10µm,andS =20 250nm2.
L
×
C
a b
R R L
V I
SQUID
V I
dc + pulse
|0/1\
Φ
1 mm dc + mw
FigureA1: a,Scanningelectronmicrograph(SEM)ofadevice(qubitandSQUIDshown)withiden-
ticaldesignparametersastheonemeasuredduringthiswork[18]. b,Schematic.Qubitloop(shaded)
andgalvanicallycoupledread-outSQUID.ThecrossesareJosephsonjunctions;R,biasresistors;C,
shuntcapacitances;L,inductances.
9
Whenthe externalmagneticflux Φ threadingthe qubitloopis close to halfa magnetic-fluxquantum,Φ /2,
0
thequbit’spotentialenergyexhibitsadouble-wellprofilewithquantizedenergylevels ¯hε/2 = I Φ , where
p b
± ±
Φ = Φ Φ /2isthefluxbiasandI = 0.18µAthepersistentcurrent. Thegroundstatesintheleftandright
b 0 p
−
potentialwellscorrespondtothediabaticstates L and R ofoppositecirculatingpersistentcurrents. Atunnel
| i | i
coupling ¯h∆/2 between the right- and left-well qubitstates opens an energygap, ¯h∆, between the groundand
excited states, 0 and 1 , at flux degeneracy, Φ = 0. The two-level Hamiltonian – analogous to a spin-1/2
b
| i | i
particleinamagneticfield–isinthe“laboratory”frame
¯h
ˆ = [(ε+δε)σˆ +(∆+δ∆)σˆ ], (7)
x z
H −2
where ¯hω = h¯√ε2+∆2 is the energy-levelsplitting; σˆ are the Paulimatrices; andδε and δ∆ are the noise
01 j
fluctuations. The quantization axis makes an angle θ = arctan(ε/∆) with σˆ (and an angle π θ with the
z
−
persistent-currenteigenstates), so that the qubit is first-orderinsensitive to flux noise when biased at the “sweet
spot” ε = 0. (We write the Hamiltonian in this way rather than with swapped x and z indices as in several
previouspapers,sothatourpulseswillbealongX andY inagreementwiththehabitualNMRlanguage.) Itcan,
alternatively,bewritteninthequbit’seigenbasis,
1
ˆ = ¯h(ω01σˆz′ +δωz′σˆz′ +δω⊥′σˆ⊥′), (8)
H −2
seeFig.A2a. Hereσˆ⊥′ denotesthatthetransversespincomponentcanincludebothσˆx′ andσˆy′. Thelongitudinal
andtransversenoises,δωz′ andδω⊥′,arefurtherdescribedbelow.
a b
Z = Z’ Z
Y = Y’ Y = Y’
D ω01 D θ Z’ ω01 div.)
/
X = X’ X ε GHz
0
1
X’ h (
/
ε = 0, ω = D ε = 4D E
01
{ X, Y, Z } is the laboratory frame
0.48 0.50 0.52
Φ /Φ
{ X’, Y’, Z’ } is the qubit’s eigenframe b 0
FigureA2: Two-levelHamiltonianindifferenteigenbases,andsimulatedenergyspectrum.
a,TheHamiltonian(7)inthe“laboratory”frame X,Y,Z hastheεfield(flux-bias)alongX,per-
{ }
pendiculartotheplaneofthequbitloop,andthefixedtunnelcoupling∆alongZ. TheHamiltonian
(8) in the qubit’seigenframe X′,Y′,Z′ makesan angle θ with Z. The figure showsbothframes
{ }
undertwobiasconditions(ε=0[θ =0◦]andε=4∆),withthelaboratoryframefixedinspace.The
twoframescoincidewhenε=0. b,Simulatedenergyspectrum.
10