Magnet field sensing beyond the standard quantum limit under the effect of decoherence

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by  Yuichiro Matsuzaki

Magnetfieldsensingbeyondthestandardquantumlimitundertheeffectofdecoherence
Yuichiro Matsuzaki,1 Simon C. Benjamin∗,1,2 and Joseph Fitzsimons2
1Department of Materials, University of Oxford, OX1 3PH, U. K.
2CentreforQuantumTechnologies,NationalUniversityofSingapore,3ScienceDrive2,Singapore117543.
Entangled states can potentially be used to outperform the standard quantum limit which every classical
sensor is bounded by. However, entangled states are very susceptible to decoherence, and so it is not clear
whetheronecanreallycreateasuperiorsensortoclassicaltechnologyviaaquantumstrategywhichissubject
totheeffectofrealisticnoise.Thispaperpresentsaninvestigationofhowaquantumsensorcomposedofmany
spinsisaffectedbyindependentdephasing. Weadoptgeneralnoisemodelsincludingnon-Markovianeffects,
andinthesenoisemodelstheperformanceofthesensordependscruciallyontheexposuretimeofthesensor
tothefield. Wehavefoundthat,bychoosinganappropriateexposuretimewithinnon-Markoviantimeregion,
anentangledsensordoesactuallybeatthestandardquantumlimit. Sinceindependentdephasingisoneofthe
mosttypicalsourcesofnoiseinmanysystems,ourresultssuggestapracticalandscalableapproachtobeating
1 thestandardquantumlimit.
1
0
2
Entanglement has proven itself to be one of the most in- limit, one can use a GHZ state, as has been demonstrated in

n triguingaspectsofquantummechanics,anditsstudyhaslead recentexperiments[9–11,13].
a toprofoundadvancesinourunderstandingofphysics. Aside Insolidstatesystems,oneofthemainbarrierstorealising
J
fromtheseconceptualadvances,theexploitationofentangle- suchsensorsisdecoherence,whichdegradesthequantumco-
3
menthasleadtoanumberoftechnicaladvancesbothincom- herenceoftheentangledstates. GHZstates,inparticular,are
1
putationandcommunication[1],andmorerecentlyinmetrol- verysusceptibletodecoherence,anddecoheremorerapidlyas

] ogy [2]. In quantum metrology, entanglement has been used thesizeofthestateincreases[14,15].Therefore,itisnotclear
h
todemonstrateenhancedaccuracybothindetectingthephase whetheraquantumstrategycanreallyoutperformanoptimal
p
- inducedbyunknownopticalelementsandforaccuratelymea- classicalstrategyundertheeffectsofarealisticnoisesource.
nt suringanunknownmagneticfield. Itisthislattercasewhich The effect of unknown but static field variations over the L
a is the focus of the present paper, and so we will adopt the spins has been studied by Jones et al [10]. The uncertainty
u terminologyoffieldsensing. of the estimation depends on the exposure time of entangled
q
Inordertoestimateanunknownfield,oneusuallyprepares statestothefield, wheretheyareaffectedbynoise, andthey
[
aprobesystemcomposedofLdistinctlocalsubsystems,ex- havefoundthat,foranoptimalexposuretimeintheirmodel,

1 poses this to the field for a certain time, and measures the thescalingoftheestimatedvalueisL−34 whichbeatsthestan-
v
probe. Comparingtheinputwiththeoutputoftheprobegives dardquantumlimit. However,theunderlyingassumptionthat
1
6 usanestimateofthefield. Importantly,thereisanuncertainty thefieldsarestaticcouldbeunrealisticformanysystems, as
5 in the estimation, and this uncertainty is related to how the actualnoiseinthelaboratorymayfluctuatewithtime. Huelga
2 probesystemisprepared. Whentheprobesystemisprepared et al have included such temporal fluctuations of the field in
√
1. inaseparablestate,theuncertaintydecreasesas1/ L[3,4] theirnoisemodel[16]andhaveshownthatGHZstatescannot
0 bythecentrallimittheorem,ascalingknownasthestandard beat the standard quantum limit under the effect of indepen-
1 quantumlimit. Ontheotherhand, bypreparingahighlyen- dent dephasing by adopting a Lindblad type master equation
1 tangled state, it is in principal possible to achieve an uncer- [16]. Even for the optimal exposure time, it was shown that
:
v taintythatscalesas1/L,knowntheHeisenberglimit[2,5]. the measurement uncertainty of a quantum strategy has the
i same scaling behavior as the standard quantum limit in their
X Mostoftheliteratureonquantumsensingfocusesonusing
noisemodel.Sinceindependentdephasingisthedominanter-
r photonstoprobeanunknownopticalelement.UsingaNOON
a state[6–8],itispossibletomeasuretheunknownphaseshift rorsourcesinmanysystems,theseresultsseemtoshowthat,
practically,itwouldbeimpossibletobeatthestandardquan-
with higher resolution than the standard quantum limit. An
tumlimitwithaquantumstrategy.
L-photonNOONstatecanachieveaphaseLtimesaslargeas
However,themodeladoptedbyHuelgaetalisaMarkovian
thaninthecaseofasinglephotonoverthesamechannel. Re-
masterequation[17]whichisvalidinlimitedcircumstances.
cent publications [9–12] have considered an analogous tech-
TheMarkovianassumptionwillbeviolatedwhenthecorrela-
niqueforfieldsensingwithspins. Inthispaper, weconsider
anexperimentinvolvingaprobeconsistingofspin-1 systems. tiontimeofthenoiseislongerthanthecharacteristictimeof
2 thesystem. Forexample,althoughaMarkovianmasterequa-
Thespinqubitscancoupletothemagneticfieldandtherefore
tion predicts an exponential decay behavior, it is known that
onecanestimatethevalueofthemagneticfieldbytheprobe.
unstable systems show a quadratic decay in the time region
Inordertoobtainhigherresolutionthanthestandardquantum
shorter than a correlation time of the noise [18, 19]. In this
paper,weadoptindependentdephasingmodelswhichinclude
non-Markovianeffectsandweinvestigatehowtheuncertainty
∗[email protected] of the estimation is affected by such noise. We have found
2
that, if the exposure time of the entangled state is within the where[σˆ ,[σˆ ,··· ,[σˆ ,ρ ]···]] denotesthen-foldedcom-
z z z 0 n
non-Markovianregion,aquantumstrategycanindeedprovide mutator of ρ with σˆ . Since all higher order cumu-
0 z
ascalingadvantageovertheoptimalclassicalstrategy. lants than the second order are zero for Gaussian noise,
LetussummarizeaquantumstrategytoobtaintheHeisen- f(t )f(t )···f(t ) can be represented by a product of cor-
1 2 n
berg limit in an ideal situation without decoherence. A state relationfunctions[20]. Therefore,thedecoherencecausedby
prepared in |+(cid:105) = √1 (|0(cid:105)+|1(cid:105)) will have a phase factor in Gaussian noise is characterized by a correlation function of
2
its non-diagonal term through being exposed in a magnetic thenoise. For Markovianwhite noise, a correlationfunction
fieldandsowehave √1 (|0(cid:105)+e−itδ|1(cid:105))whereδ denotesthe becomesadeltafunctionwhileacorrelationfunctionbecomes
2
detuning between the magnetic field and the atomic transi- a constant for non-Markovian noise with an infinite correla-
tion. On the other hand, when one prepares a GHZ state tiontimesuchas1/f noise. Toincludebothnoisemodelsas
|ψ(cid:105)GHZ = √12(|00···0(cid:105)+|11···1(cid:105)) and exposes this state specialcases,weassumethecorrelationfunction
tothefieldforatimet,thephasefactorisamplifiedlinearly
asthesizeofthestateincreasesas f(t1)f(t2)= √2πe−|t1−τc2t2|2, (6)
1
|ψ(t)(cid:105)= √ (|00···0(cid:105)+e−iLtδ|11···1(cid:105)). (1)
2 whereτ denotesthecorrelationtimeofthenoise. Inthelimit
c
Therefore,theprobabilityoffindingtheinitialGHZstateafter τ → 0, this correlation function becomes a delta function,
c
atimetisgivenbyP = 1+1cos(Ltδ).Inpractice,onemay whileinthelimitofτ →∞itbecomesconstant.
2 2 c
usecontrol-notoperationstomaptheaccumulatedphasetoa Sincef(t )f(t )···f(t )canberepresentedbyaproduct
1 2 n
singlespinforaconvenientmeasurement[10]. Thevariance ofcorrelationfunctions,weobtain
oftheestimatedvalueisthengivenby
∆2δ = P(|d1P−/dPδ)|/2N (2) ρI(t) = (cid:88)∞ (−41tnγ!(t))n[σˆz,[σˆz,··· ,[σˆz,ρ0]···]]2n
n=0
whereN isthenumberofexperimentsperformedinthisset- = (cid:88) e−14|s−s(cid:48)|2γ(t)t|s(cid:105)(cid:104)s|ρ0|s(cid:48)(cid:105)(cid:104)s(cid:48)| (7)
ting[16]. ForagivenfixedtimeT,onecanperformthisex-
s,s(cid:48)=±1
periment T/t times where t is the exposure time, and so we
have N = T/t. Hence we obtain |∆2δ| = 1 and so the where|s(cid:105)isaneigenvectorofσˆ . Also,γ(t)denotesthesin-
TL2t z
uncertaintyinδscalesasL−1,theHeisenberglimit. glequbitdecoherenceratedefinedas
First let us consider the decoherence of a single qubit.
Lreaptreers,ewnteswrainlldogmenecrlaalsisziecatlofiGelHdsZtsotaitnedsu.ceOduerpnhoaissiengm.oWdeel γ(t)= 4λ2τc2(−√1π+t e−τt2c2) +4λ2τcerf(τt ) (8)
consideraninteractionHamiltoniantodenoteacouplingwith c
anenvironmentsuchas
whereerf(x)istheerrorfunction. Notethat,fort (cid:29) τ ,the
c
HI =λf(t)σˆz (3) decoherence rate becomes constant as γ(t) (cid:39) 4λ2τc. So, in
thisregime,wecanderiveaMarkovianmasterequationfrom
where f(t) is classical normalized Gaussian noise, σˆ is a
z (7),whichhasthesameformasadoptedbyHuelgaetal[16].
Pauli operator of the system, and λ denotes a coupling con-
stant. Also, we assume symmetric noise to satisfy f(t) = 0
dρ (t)
where this over-line denotes the average over the ensemble dIt (cid:39)−λ2τc[σˆz[σˆz,ρI(t)]] (t(cid:29)τc) (9)
of the noise. When we solve the Schro¨dinger equation in an
interactionpicture,weobtainthefollowingstandardform Notethat,althoughourmodelcanbeapproximatedbyMarko-
(cid:88)∞ (cid:90) t (cid:90) t1 (cid:90) tn−1 vian noise in the long time limit (t (cid:29) τc), we are interested
ρ (t)−ρ = (−iλ)n dt dt ··· dt inthetimeperiodst ∼ τ andt (cid:28) τ wherenon-Markovian
I 0 1 2 n c c
n=1 0 0 0 effectsbecomerelevant.
[H (t ),[H (t ),··· ,[H (t ),ρ ]···]] (4) For an initial state |+(cid:105) = √1 (|0(cid:105) + |1(cid:105)), the non-
I 1 I 2 I n 0 2
diagonal terms of the density matrix show a decay behavior
where ρ = |ψ(cid:105)(cid:104)ψ| is an initial state and ρ (t) is a state in
0 I of(cid:104)0|ρ (t)|1(cid:105) = 1exp[−γ(t)t]. Fort (cid:29) τ , thestateshows
theinteractionpicture. Throughoutthispaper,werestrictour- I 2 c
selves to the case where the system Hamiltonian commutes anexponentialdecay(cid:104)0|ρI(t)|1(cid:105) (cid:39) 12e−4λ2τct. Ontheother
with the operator of noise, as this constitutes purely dephas- hand,wehave(cid:104)0|ρI(t)|1(cid:105)(cid:39) 12e−√4πλ2t2 (quadraticdecaybe-
ing noise. By taking the average over the ensemble of the havior) for t (cid:28) τ , which is the typical decay behavior of
c
noise,weobtain 1/f noise [21–23]. The behavior of the decoherence rate is
(cid:88)∞ 1 (cid:90) t (cid:90) t (cid:90) t illustratedinFig.1.
ρI(t)−ρ0 = n!(−iλ)n dt1 dt2··· dtn We next consider decoherence of a GHZ state induced by
n=1 0 0 0 randomclassicalfields. ExtendingtheHamiltonianin(3),the
f(t )f(t )···f(t )[σˆ ,[σˆ ,··· ,[σˆ ,ρ ]···]] (5) interactionHamiltoniandenotingrandomclassicalfieldsfora
1 2 n z z z 0 n
3
choiceoftforwhichwebeatthestandardquantumlimit. We
findthatthislimitcanindeedbebeatenprovidedthatwechose
shortertvaluesforlargersystems. Forexample,supposethat
wechosetaccordingtotherulet = sL−z,wheresisacon-
stantwiththedimensionoftime,andz isanon-negativereal
numberwhoseoptimalvaluewewilldetermine. Then, from
(8),wehave
4sλ2 12s2λ2/τ
|γ(t)− √ |≤ √ c (14)
πLz πLz(L2z−s2/τ2)
c
for large L and hence the decoherence rate scales as γ(t) =
FIG.1: Wehaveplottedthebehaviorofasinglequbitdecoherence
Θ(L−z). Throughoutthispaper,forafunctionf(L),wesay
ratefor1/f noise √4πλ2t(thegreenline), Markoviannoise4λ2τc f(L)=Θ(Ln)ifthereexistspositiveconstantsJ andKsuch
(theredline),andourclassicalnoisemodel(bluedottedline)γ(t)
thatJLn ≤f(L)≤KLnissatisfiedforallL.
definedin(8)whereλandτ denoteacouplingconstantwiththeen-
c
vironmentandacorrelationtimeofthenoise,respectively.ForanL- In(13),thetermintheexponentialγ(t)tLgoestoinfinity
qubitGHZstate,thedecoherenceratebecomesLtimesgreaterthan as L → ∞ for z < 1/2, and so the uncertainty |∆2δ| di-
thevalueplottedhere. Notethatournoisemodelshowsatransition verges,whichmeansthatalargeGHZstatebecomesuseless
fromanon-MarkovianquadraticdecaytoanexponentialMarkovian toestimateδ. Therefore,weconsiderthecaseofz ≥1/2. By
decay.Here,wefixedparametersasλ=0.25andτc =1. performingaTaylorexpansionofe2γ(t)Lt,weobtain
e2γ(t)Lt−1 (cid:88)∞ (2γ(t)Lt)n
many-qubitsystemisasfollows: = =Θ(Lz−3) (15)
L4t3 L4t3·n!
n=1
L
(cid:88)
HI =λ fl(t)σˆz(l) (10) forz ≥1/2whereweuse(14).
l=1 So,from(13)and(15),weobtain
wheref (t)denotesthenoiseactingatsitelandhasthesame
l ∆2δ =Θ(L−2+z) (16)
characteristicsasf(t)mentionedabove. Also,sincewecon-
sider independent noise, we have fl(t)fl(cid:48)(t(cid:48)) ∝ δl,l(cid:48). When for z ≥ 1. Therefore, when z = 1, we achieve a scaling of
2 2
we let a GHZ state be exposed to a magnetic field under the the uncertainty |∆δ| = Θ(L−34) and this actually beats the
effectoftherandommagneticfields, thestatewillremainin
standardquantumlimit. Notethatthisisthesamescalingas
thesubspacespannedby(cid:78)L |0(cid:105) and(cid:78)L |1(cid:105) becausewe
l=1 l l=1 l themagneticsensorundertheeffectofunknownstaticfields
areonlyconsideringphasenoise. Thus,wecanusethesame
studiedin[10],andsoourresultforthefluctuatingnoisewith
analysisasforasinglequbit. InthetheSchro¨dingerpicture,
timebecomesanaturalgeneralizationoftheirwork.
wewillobtain
Thedecoherencemodeldescribedaboveisaclassicalone.
L L Next, we make use of a quantized model where the environ-
ρ(t)= 12(cid:0)(cid:79)|0(cid:105)l(cid:104)0|(cid:1)+ 12eiLtδ−Lγ(t)t(cid:0)(cid:79)|0(cid:105)l(cid:104)1|(cid:1) mentismodeledasacontinuumoffieldmodes. TheHamil-
l=1 l=1 tonianofthesystemandtheenvironmentaredefinedas
L L
+ 21e−iLtδ−Lγ(t)t(cid:0)(cid:79)|1(cid:105)l(cid:104)0|(cid:1)+ 12(cid:0)(cid:79)|1(cid:105)l(cid:104)1|(cid:1) (11) H = δ (cid:88)L σˆ (l)+(cid:88)ω ˆb† ˆb +(cid:88)g σˆ (l)ˆb† ˆb (17)
l=1 l=1 2 z k l,k l,k k z l,k l,k
l=1 l,k l,k
whereδisthephaseinducedbythemagneticfieldtobemea-
sured. It is worth mentioning that the decoherence rate for whereˆb andˆb† denoteannihilationandcreationoperators
l,k l,k
a L-qubit GHZ state becomes L times the decoherence rate for the bosonic field at a site l. Also, since we consider in-
γ(t)ofasinglequbit.Therefore,thevarianceoftheestimated dependentnoise,weassumethatˆb commuteswithˆb† for
l,k l(cid:48),k
valuebecomes l(cid:54)=l(cid:48). ThismodelhasbeensolvedanalyticallybyPalmaetal
e2γ(t)Lt−1 1 [24]andthetimeevolutionofaGHZstateisgivenby
∆2δ = + (12)
TL2tsin2(Ltδ) TL2t L L
ρ(t)= 1(cid:0)(cid:79)|0(cid:105) (cid:104)0|(cid:1)+ 1eiLtδ−LΓ(t)t(cid:0)(cid:79)|0(cid:105) (cid:104)1|(cid:1)
where we use (2), and therefore we obtain the following in- 2 l 2 l
equalitybyusingx≥sinx≥ 2xfor0≤x≤ π: l=1 l=1
π 2 L L
e2γ(t)Lt−1 ≤(∆2δ)− 1 ≤ π2(e2γ(t)Lt−1) (13) + 12eiLtδ−LΓ(t)t(cid:0)(cid:79)|1(cid:105)l(cid:104)0|(cid:1)+ 12(cid:0)(cid:79)|1(cid:105)l(cid:104)1|(cid:1), (18)
TL4t3δ2 TL2t 4TL4t3δ2 l=1 l=1
These inequalities depend on both the system size L and where Γ(t) denotes a decoherence rate defined as Γ(t) =
thechoiceofexposuretimet. Wewishtoseeifthereisany 1log(1+ω2t2)+ 2log(sinh(πTt))wherewehavetakenthe
t c t πTt
4
Boltzmann constant k = 1. Here, T and ω denote the wouldbesuitableforthefirstexperimentaldemonstrationof
B c
temperature and cut off frequency respectively. As we take our model. For example, it is known that nuclear spins of
k =1,thetemperatureT hasthesamedimensionasthefre- donor atoms in doped silicon devices, which have been pro-
B
quencyω . WehaveaconstantdecoherencerateΓ(t)(cid:39)2πT posed as qubits for quantum computation [26], are dephased
c
for t (cid:29) T−1, which signals Markovian exponential decay, mainlyby1/fnoise[26,27]andsotheymayprovesuitableto
whilewehaveΓ(t)(cid:39)ω2tfort(cid:28)ω−1,whichisthecharac- demonstrateourprediction.
c c
teristic decay of 1/f noise. So, by taking these limits, this
Inconclusion,wehaveshownthat,undertheeffectofinde-
model can also encompass both Markovian noise and 1/f
pendent dephasing, one can obtain a magnetic sensor whose
noise. Giventhecalculationsintheprevioussection,thevari- uncertainty scales as Θ(L−34) and therefore beats the stan-
anceoftheestimatedvalueforthemagneticfieldbecomes dard quantum limit of L−21. We determine that, to outper-
form a classical strategy, the exposure time of the entangled
(1+ω2t2)2L(sinh(πTt))4L−1 1
∆2δ = c πTt + . (19) states to the field should be within the non-Markovian time
TL2t·sin2(Ltδ) TL2t region where the decoherence behavior doesn’t show expo-
nentialdecay. Sincethenoisemodelsadoptedherearequite
Byperformingacalculationexactlyanalogoustothecaseof
general,ourresultssuggestascalablemethodtobeatthestan-
the random classical field considered above, one can show
dardquantumlimitinarealisticsetting.
that the scaling law for the uncertainty becomes ∆2δ =
Θ(L−2+z) for z ≥ 1 when we take an exposure time t = TheauthorsthankM.SchaffryandE.Gaugerforusefuldis-
2
s/Lz. On the other hand, the uncertainty will diverge as L cussions. ThisresearchissupportedbytheNationalResearch
increasesforz < 1. Therefore, bytakingz = 1, theuncer- FoundationandMinistryofEducation,Singapore.YMissup-
2 2
tainty scales as Θ(L−34) and so one can again beat the stan- portedbytheJapaneseMinistryofEducation,Culture,Sports,
ScienceandTechnology.
dardquantumlimitasbefore.
Wenowprovideanintuitivereasonwhytheuncertaintyof
theestimationdivergesforlargeLwhenz isbelow 1 inboth
2
ofnoisemodels. Ithasbeenshownthatanunstablestateal-
ways shows a quadratic decay behavior in an initial time re-
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