On-Line Estimation of Local Oscillator Noise and Optimisation of Servo Parameters in Atomic Clocks

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by  Ian D. Leroux

On-LineEstimationofLONoiseandOptimisationofServoParametersinAtomicClocks
∗
IanD.Leroux, NilsScharnhorst,StephanHannig,JohannesKramer,LennartPelzer,MariiaStepanova,andPietO.Schmidt
QUESTInstituteforExperimentalQuantumMetrology,Physikalisch-TechnischeBundesanstalt,38116Braunschweig,Germany
(Dated:25thJanuary2017)
Foratomicfrequencystandardsinwhichfluctuationsofthelocaloscillatorfrequencyarethedominant
noisesource,weexaminetheroleofthetheservoalgorithmthatpredictsandcorrectsthesefrequencyfluc-
tuations. Wederivetheoptimallinearpredictionalgorithm,aswellaspracticalproceduresforoptimising
theparametersofaconventionalintegratingservoandcharacterisingtheflickerandrandom-walknoise
ofthelocaloscillatorwithoutinterruptingthenormaloperationofthestandard. Inparticular, weshow
thatnear-optimalservoparameterscanbecalculatedwithoutindependentknowledgeofthelocal-oscillator
noise,basedonlyonmonitoringoftheerrorsignal.Usingsimpleanalyticalmodelsandnumericalsimula-
tions,weestablishoptimumprobetimesasafunctionofclockatomnumberandofthedominantnoisetype
inthelocaloscillator. Thisleadstoanargumentforthesuperiorityofmethodsthatincreasethemeasure-
mentsignal-to-noiseratio(increasingatomnumber,spinsqueezing)overthosethatincreasetheeffective
transitionfrequency(maximally-correlatedstates)whentryingtoimprovethestabilityoflocal-oscillator-
7 limitedfrequencystandards.
1
0
2 The instability of frequency standards limits the total termfractionalinstability[11]
n uncertainty achievable in a measurement of finite dura-
(cid:115)
a tion[1,2]. Thislimitcanbepracticallyrelevantevenwhen 1 T
J performingmeasurementsofstaticfrequencyratios,since σC(τ)=ωT(cid:112)N τc (2)
4 many-month-long measurement campaigns place strin-
2 gent demands on the reliability of all components in an where T is the duration of a single Ramsey interrogation

experiment. Instability becomes a fundamental concern and T is the length of the frequency standard’s operating
] c
h whenattemptingtomeasuretime-varyingfrequencyratios. cycle, such that τ/T measurements can be performed in
c
p For instance, in the emerging field of chronometric level- an averaging time τ. This quantum projection noise limit
- ing[3–5],directobservationoftidalfluctuationsexpectedin (QPN)1forclocksusinguncorrelatedatomsdependsonthe
m
thegravitationalredshift[6]requiresfrequencyratiomeas- experimenter’schoiceofprobetimeT,becomingarbitrarily
o urementswithafractionaluncertaintyatthelevelof10−18 smallforsufficientlylongprobetimes. Thus,Eqn.2setsno
t
a tobecompletedinamatterofhours. Physicsbeyondthe limitonachievableclockinstabilityatlongaveragingtimes
s. StandardModelmightbedetectableinclockfrequencyra- unless some additional scale in the problem restricts the
c tiomeasurementsaspostulatedtransientshiftsassociated choiceofT.
si withdark-matterdomainwalls[7]orultralightscalardark- One such restriction is set by excited-state decay in the
y mattercandidates[8,9].Searchesforsuchsignalswouldre- atoms, which sets a fundamental limit to interrogation
h quireevenfastermeasurements,completedinseconds. At times. Theperformanceofopticalfrequencystandardsop-
p suchtimescales,thestatisticaluncertaintyduetoinstabil- eratingatthislimithasbeenanalysedinRefs.[12–15].How-
[
ityplaysafargreaterrolethanlong-termsystematicuncer- ever, formanyoftheopticalfrequencystandardsnowbe-

1 taintyincurrentstandards. ing investigated, frequency fluctuations of the local oscil-
v Of the noise processes contributing to the instability of latorrestrictT tolessthanasecondevenwhentheatoms’
7 atomicfrequencystandards, themostfundamentaloneis excited-statelifetimeismeasuredinminutes(87Sr)oreven
9
quantum projection noise [10], which arises from the dis- years (171Yb+) [1]. Because the local oscillator’s noise is
6
cretenessinthemeasurementresultsobtainablefromafi- common to all the atoms in the standard, and because
6
0 nitenumberofatoms. ForanensembleofN uncorrelated it typically exhibits significant power-law temporal correl-
. two-levelatoms, thisnoiseimposesaminimumstatistical ations, its effects are qualitatively different from those of
1
uncertainty excited-state decay. In fact, it might at first glance seem
0
7 oddthatlocal-oscillatornoiselimitsclockstabilityatall:the
1
1 ∆φ =(cid:112) (1) localoscillatorfrequencyisinsomesensethemeasurandin
QPN
: N anatomicfrequencystandardanditsfluctuationsarecon-
v
stantlymonitoredandcorrected. Local-oscillatornoiseaf-
i
X on any measurement of the phase accumulated in an fectsthestabilityofthestandardonlytotheextentthatit
r atomic superposition state. For a standard operating at a cannotbecorrectedbyfeedbackfromtheatoms. Thiscan
a frequencyωandintheidealcaseofdead-time-freeRamsey happen,forinstance,ifthecyclicatomicinterrogationpro-
interrogationwithouttechnicalnoise, thisleadstoalong- tocol allows undetected aliased frequency components of
∗[email protected] 1sometimesreferredtoastheStandardQuantumLimit(SQL)
2
thelocal-oscillatornoisespectrumtocontaminatetheout- x x−h
LO Output
put signal of the standard, a phenomenon known as the
Dick effect [16]. Even in the absence of the Dick effect,
h
however,thequantisedmeasurementsignalfromtheatoms
hasafundamentallylimiteddynamicrange:onecannotex-
ω
tract more than log2(N+1) bits of frequency information Servo e=y−h≈x−h ?
from a single measurement of N atoms [17]. The useful Reference
domain of the measurement, i.e. the frequency band in
whichitcanbeunambiguouslyinterpreted,mustbebroad
enoughto cover the frequencies whichthe local oscillator Figure1. Generalmodelofaperiodically-stabilisedatomicclock.
islikelytoemitintheinterrogation. Frequencyexcursions Alocaloscillator(LO)emitsasignalwithafractionaldeviationx
fromthenominalclockfrequency. Theservocontrollerattempts
beyond the domain for which the reference provides use-
topredict x andcorrectsthefrequencybyitspredictionh. The
fulinformationleadtolessinformativemeasurementres-
correctedsignal(withfractionalfrequencydeviationx−h)isthen
ults,andhencetodegradedinstability.Intheworstcasethe
usedtointerrogateanatomicreference,whichproducesanestim-
servo,workingfromambiguousoruninformativemeasure-
atee ofthepredictionerroror,equivalently,anestimatey ofthe
mentresults,maybeunabletokeeptheoutputfrequency LO’sunknownfrequencydeviationx.Theseestimatescanbeused
lockedtotheatomicreference. Theoutputfrequencythen bytheservoinfuturepredictions.
eitherhopsbetweendifferentzero-crossingsofafrequency-
periodicRamseyerrorsignalordriftsaimlesslyfarfromthe
resonanceofaRabierrorsignal. Thiscaseiscatastrophic tionVIconsidersthemeritsofusingentangledatomicstates
and the operating parameters must be chosen to make it to modify the phase resolution of Eqn. 1, giving a simple
vanishinglyunlikely. Thus,evenintheabsenceoftheDick dimensional argument for the disappointing performance
effect(e.g. withdead-time-freeRamseyinterrogation[18]), ofmaximally-correlatedstatesinatomicclocksandarguing
theachievablemeasurementresolutionultimatelydepends forthesuperiorityofstatesthatenhancethedynamicrange
onthescaleoflocal-oscillatorfrequencyfluctuationsseen ofatomicmeasurements[17,24–26]. Thisresultiscomple-
bytheatoms,andhenceontheperformanceoftheclock’s mentarytothatofRef.[12],whichconsideredindependent
feedbackloopwhichcorrectsthesefluctuations. dephasingoftheatomsratherthanthecollectivedephas-
ingassociatedwiththeLO,andtakesintoaccounttemporal
In this work, we study the limits to the stability of fre-
correlations intheLOnoise rather thanassumingawhite
quencystandardsdominatedbylocal-oscillatornoisewith
spectrum as in Ref. [20, 27]. Section VII considers the in-
realistic temporal correlations. We focus on clocks using
stabilityoftheclockatshorttimes,whichmaybelimitedby
asingleensembleofatomsperiodicallyinterrogatedusing
finitefeedbackgainratherthanmeasurementnoise,show-
thesameprotocolforeveryinterrogationcycle, whosein-
ingthatthesecondintegratorrecommendedinRef.[15]to
stabilitywequantifyusingtheAllanvarianceatlongtimes.
correctforlineardriftoftheLOisalsonecessarytosatur-
Our work is thus less general, but more directly relevant
atetheQPNlimitinthepresenceofrandom-walknoise.We
to current experiments, than analyses of multi-ensemble
concludewithsomeremarksonproposedfrequencystand-
clocks or of interrogation protocols which are modified
ardswhosedesigndoesnotfollowtheconventionalpattern
on-the-fly [19–22], and our approach is a more concrete
weconsiderinthiswork.
complement to the derivation of universal performance
bounds in mathematically idealised settings [22, 23]. Us-
ingsimpleanalyticalargumentsandnumericalsimulations
of clocks with different local-oscillator noise spectra, we I. SETUPANDNOTATION
study the performance of the servo controller which pre-
dicts and corrects local-oscillator noise and then analyse Figure1sketchesthestructureofthefrequencystandards
its implications. After establishing notation and conven- we consider and summarises the notation we will use for
tionsinSec.I,webeginbyderivingtheoptimallinearpre- thevarioussignalswemustconsider.Thegoalofthestand-
dictionalgorithmandevaluatingitsperformanceinSec.II. ard is to produce a continuous classical oscillatory signal
In Sec. III we show that a feedback controller with near- whosefrequencycorrespondstoareferencetransitionfre-
optimalperformancecanbedesignedwithoutpriorknow- quencyωinaparticularatomicspecies.Astheclassicalsig-
ledgeofthenoisespectrum,bymonitoringtheerrorsignals nalisgeneratedbyamacroscopiclocaloscillator(LO)sub-
innormalclockoperation.Wealsoshowthatthesametech- ject to environmental perturbations, its frequency ω will
L
niques provide useful diagnostic information on the local differfromthetargetfrequencyωbyafluctuatingfractional
oscillator’s noise, allowing on-line monitoring of its per- discrepancyx=(ω /ω)−1. Thescaleofthesefluctuations
L
formance. We turn to the effects of the noise in Sec. IV, issummarisedintheAllandeviationσ (τ)oftheLO.Inor-
L
in which we derive a modification to the QPN that takes dertosuppressthesefluctuations,aservocontrollergener-
intoaccounttheperformanceoftheservocontroller. This atesapredictionhoftheLOfrequencyerror,whichisused
modified QPN formula predicts an overall limit to achiev- tofrequency-shifttheLOoutputsignalbacktoatomicres-
ableclockinstability,whichisattainedforanoptimalchoice onance. Theresultingsignal,withnetfractionalfrequency
ofatomicinterrogationtimethatwediscussinSec.V.Sec- errorx−h,isprovidedtotheusersofthestandardandhas
3
anAllandeviationσ (τ). Thiscorrectedsignalissupplied power-lawnoise,suchthatσ2(τ)∝τµ,withµ=−1,0,1for
C L
toareference,whereitinteractswithN atomsaccordingto whitefrequencynoise,flickerfrequencynoiseandrandom
somefixedinterrogationprotocol,suchasRamseyorRabi walkoffrequencynoise, respectively. Asarguedinthein-
interrogation. Measurementoftheatoms’stateattheend troduction,theLOnoisegivestheproblemacharacteristic
oftheinterrogationprotocolconveyssomeinformationon time scale which ultimately limits the useful resolution of
theresidualfrequencyerror x−h,whichweexpressasan measurementsontheatoms.WedefinethistimeZ,without
error estimate e. The error estimate might, for instance, assumingaparticularformofLOnoisespectrum,bytheim-
correspondtotheimbalanceofatomicstatepopulationsat plicitequation
theendofaRamseysequencedividedbytheaccumulated
phase ωT. For consistency, we express the error estimate σL(Zc)ωZ=1rad, (3)
inthesameunitsasx andh,sothat y =h+e isanestim-
ateoftheLO’suncorrectedfrequencyerrorx,onethatuses whereZcisthecycletimeoftheclockwhenoperatedwith
onlythemostrecentatomicdataandtakesnoaccountof a probe time Z. In other words, Z is the choice of probe
previousmeasurements. Notethaty andedifferfrom,and timeforwhichtheLOAllandeviationatoneclockcycleis
fluctuatemorethan,x andx−hrespectively,becausethey as large as the quantum projection noise of a single atom
areaffectedbythenoiseoftheatomicreference. (Eqn. 2 with N = 1). This definition lets us combine the
We consider periodically-stabilised frequency standards LOnoiseandthechoiceofprobetimeintoasingledimen-
with an operating cycle of period T , where the reference sionless parameter T/Z which can be compared between
c
providesaseriesoferrorestimates{e }. Atanygivenpoint clocksofdifferenttypesusingLOswithdifferentperform-
i
intime,welabelthemasfollows:e isthemostrecentavail- ance.NotethatZ willbeontheorderofafewsecondsfora
1
ableerrorestimate,e theprecedingerrorestimate,andso typicalcurrentopticalfrequencystandardwithafractional
2
forth. e isthentheerrorestimatewhichwillbeproduced LOinstabilityaround10−16.
0
attheendofthecurrentoperatingcycle.Welabeltheother
signalssimilarly:h istheservo’spredictionofthe(average)
0
LO frequency error x0 during the current operating cycle, II. SERVOCONTROLLERDESIGN
while h and x correspond to the jth most recent com-
j j
pletedcycle. Causalityrequiresthattheservocomputethe
Wenowfocusourattentionontheservo.Givenahistory
predictionh withoutknowledgeofe ,usingonly{e ,e ,...}
0 0 1 2 ...,h ,h ,h of its own past predictions and of the corres-
3 2 1
or,equivalently,{y ,y ,...}.
1 2 pondingerrorsignals...,e ,e ,e obtainedfromtheatomic
3 2 1
The Allan deviation σC(τ) is that of the physical signal reference,itmustmakeapredictionh0oftheLOfrequency
producedbythefrequencystandardasitisoperating.With inthenextoperatingcycle.Thepredictionshouldtakeinto
the exception of Sec. VII, most of the analysis presented account the temporal correlations of the LO noise, which
in this work also applies to “paper clocks”, i.e. virtual sig- dictatethetimescaleoverwhichpastmeasurementresults
nalsgeneratedbypost-processingmeasurementdata. Al- remainrelevanttopredictingfutureLObehaviour.
thoughthepost-processingneednotrespectcausalityand Webeginbyconsideringthesimpleintegrator,thebasic
canuselatermeasurementstocorrectestimatesofthefre- buildingblockoftheservoalgorithmusedinmostcontem-
quency at earlier times, the quality of the measurements poraryopticalfrequencystandards[15].Inournotation,the
themselves still depends on the ability of the (causality- simpleintegratormakestheprediction
respecting)servotokeepthecorrectedLOfrequencynear
the atomic resonance frequency ω while the clock is run- h =h +ge (4)
0 1 1
ning,andconstraintsonthisabilityaffecttheperformance
of the reference no matter how the resulting data is sub- where g is a dimensionless gain specifying the fraction of
sequentlyused. thefrequencyerrormeasuredinthelastcycletoapplyasa
Whereitisnecessarytoassumeadefiniteinterrogation correctiontothelastprediction.Thepredictioncanalsobe
protocol in the atomic reference, we will focus on dead- expressed in terms of past estimates of the LO’s fractional
time-freeRamseyinterrogation,wherethemeasuredsignal frequencydeviation{y }asfollows:
k
depends on the average of the corrected signal frequency
∞
duringsomeinterrogationtimeT. Whileweassumeinour h = (cid:88)g(1−g)k−1y . (5)
examplesthatT,whichsetsthefrequencyresolutionofthe 0 k
k=1
interrogation,isequaltoT ,whichsetstherepetitionrateof
c
theinterrogationcycle,thetwotimesareconceptuallydis- While Eqn. 4 is easier to implement, Eqn. 5 is easier to
tinct and we will use separate symbols for them through- reasonaboutbecausethestatisticalpropertiesoftheestim-
out. Referenceswhoseoperatingcycleincludesdeadtime atedLOfrequencyyaremostlydeterminedbytheLOnoise
(T < Tc) or which use a different interrogation protocol andbythemeasurementnoiseofthereference,depending
(suchasRabiorhyper-Ramsey[28,29])willsufferfromthe onlyweaklyonthedesignoftheservocontrolleritself.Toa
Dickeffect,whichcanbemodelledasadditionalmeasure- goodfirstapproximation,then,wecantakethefluctuations
mentnoiseintheatomicreference. andcorrelationsofthe{y }asgiven,andtrytochooseg so
k
InnumericalexampleswewillconsiderLOswithsimple astominimisetheerrorofthepredictionh .
0
4
It is instructive to study the broader class of linear pre- Althoughlinearpredictorswitharbitrarycoefficientsare
dictors,whosepredictionsareweightedaveragesofpastLO not difficult to implement following Eqn. 6, one can also
frequencyestimatesoftheform use the preceding formalism to optimise the gain of con-
ventionalintegrators. AppendixBderivesanexplicit,albeit
h0=(cid:88)wkyk, (6) cumbersome,formulafortheoptimalintegratorgaingiven
k knownnoisemodelparameters. Alternatively,onecanuse
wheretheweightsw arerequiredtosatisfythenormalisa- Eqn.11tochooseavectorofweightsforahypotheticallin-
k
tioncondition earpredictorandthensimplysettheintegratorgaintothe
leadingentryofthisvectorg =w . Simulationsshowthat
(cid:88)wk=1. (7) for common power-law noise pro1cesses, the resulting in-
k tegratingservoperformsalmostaswellastheoptimallin-
earpredictor,withapenaltyoflessthan10%inthepredic-
ThesimpleintegratorofEqn.5isaspecialcaseofalinear
predictor,withw =g(1−g)k−1. Theoptimisationofsuch tionvariance. Theformalismwehavedevelopedcanthus
k
beusedtooptimisetheparametersofaconventionalinteg-
linearpredictorshasbeenstudiedextensivelysincethepi-
ratingservoalgorithm,withoutrequiringanymodifications
oneeringworkofWiener[30]andKolmogorov[31](seee.g.
toanalready-runningclockexperiment.Aswewillsee,this
Ref. [32]). Here we derive the minimum-mean-squared-
optimisation can be performed even without prior know-
error predictor in a form similar to that used for ordinary
ledgeoftheexperiment’snoisecharacteristics.
krigingingeostatistics(seee.g. [33]). Webeginbycomput-
We quantify the scale of theservo’s predictionerrors by
ingthemeansquareddifferencebetweenthepredictionh
0
thedimensionlessvariance
andthenextfrequencyestimatey :
0
(cid:42) (cid:43) v=(cid:173)((x−h)ωT)2(cid:174) (12)
〈(h −y )2〉= (cid:88)(cid:88)w w (y −y )(y −y ) (8)
0 0 j k j 0 k 0
j k of the phase accumulated in the Ramsey interrogation of
=w(cid:214)Cw, (9) durationT. Thevariancev playsanimportantroleinde-
terminingboththerobustnessofthelocktoatomicreson-
where we have collected the weights {w } into a vector w anceandtheachievablelong-termstability(seeSec.IV),so
k
and introduced the two-sample covariance matrix for the that it is worth studying its behaviour. The solid lines in
estimatedLOfrequency,whoseentriesaredefinedas Fig.2illustratetheperformanceoflinearpredictors,based
onsimulationsofclockoperationwithbetween1and104
Cjk=〈(yj−y0)(yk−y0)〉. (10) atomsinthereference.Asafunctionofthechoiceofprobe
timeT/Z,andthusoftheratiobetweenLOnoiseandmeas-
Notethat〈(h −y )2〉isnotthesameasthemeansquared
0 0 urementnoise, onecandistinguishthreequalitativelydif-
prediction error 〈(h −x )2〉, since it also includes the
0 0 ferentregimes.Inthelimitoflargeatomnumbersandlong
noise of the atomic reference which estimates that error.
probetimes, thesimulationsapproachan N-independent
Provided, however, that the atomic reference is unbiased,
limit
thesamechoiceofweightswillminimiseeithermeasureof
noise, soweproceedtominimiseEqn.9andfindthatthe (cid:181)T(cid:182)2+µ
v=ξ . (13)
optimumweightssatisfy Z
1 This is the scaling one expects for the case where the LO
1 noisecompletelydominatesthemeasurementnoiseofthe
 
Cw=λ1 (11) reference,giventhepostulatedpower-lawscalingoftheLO
1 noisewithexponentµ(c.f. Sec.I)andthedefinitionof Z
.. in Eqn. 3. The proportionality constant ξ can be estim-
.
ated from the simulations or derived using the formulae
withλaLagrangemultiplierthatmustbechosentosatisfy in Appendix A, and varies between 1 (for a white-noise-
dominatedLO)and2(forarandom-walk-noise-dominated
the normalisation constraint of Eqn. 7. Thus the optimal
weightscanbefoundbysolvingEqn.11forw/λandnor- LO). In the opposite limit of low atom number or short
probe time, the (white)quantumprojectionnoisedomin-
malisingtheresult,providedthatoneknowsthecovariance
atesandtheservoperformancedependsonlyonthenum-
matrixC. Ifthenoisepropertiesofthecomponentsinthe
berofmeasurementsnwhichitaveragesinmakingitspre-
frequency standard are known, then C can be computed
diction,andthus
simplyasthesumofmatricesforeachindependentnoise
process. Explicit expressions for the C matrix associated
1
withaknownnoisespectrumareprovidedinAppendixA. v≈ (14)
Nn
As discussed in Sec. III,C can also be estimated, and the
>
servocontrolleroptimised,without priorknowledgeofthe for sufficiently short probe times. For n ∼ 20 this limit
systemnoiseproperties, usingonlydatageneratedduring hasnoimpactoncorrectlyoptimisedclockoperation(see
normalclockoperation. Sec.V).Betweenthetwolimitsconsideredabovethereisa
5
So far, we have discussed the simple integrator and its
(a)White(µ=−1)
generalisations. Practical frequency standards, however,
1100−−11
mustuseadouble-integratortocorrectforsteadydriftsin
theLOfrequency[15]. Withtheadditionofthesecondin-
1100−−22 tegrator,theservopredictionsbecome
1100−−33 h0=h1+ge1+g2(cid:88)ek (16)
k
1100−−44
or
(b)Flicker(µ=0) =(cid:88)wkyk+g2(cid:88)ek. (17)
1100−−11 k k
Asidefromitsroleinsuppressingsteady-statefrequencyer-
1100−−22 rors with drifting LOs, the second integrator is necessary
for the servo to have enough low-frequency gain to attain
v
1100−−33 theprojectionnoiselimitinmany-atomclocks, apointto
which we will return in Sec. VII. However, as long as its
1100−−44 gain g2 is chosen low enough to avoid servo oscillations,
the additional integrator has only a negligible impact on
the variance of the prediction errors2. The controller can
(c)Random-Walk(µ=+1) thusbedesignedby optimisingasimple integrator or lin-
10−1 earpredictorasdiscussedabove,andthenaddingthedrift-
correctionintegratorwithagaing (cid:191)w =g.
2 1
Besides minimising prediction variance, another desir-
10−2
ablefeatureinpracticalservocontrollersisrobustness,the
quality of remaining locked to the (correct) atomic reson-
10−3 anceforlongperiods.Inprinciplethetwoqualitiesaredis-
tinct,butwefindempiricallythatforwell-optimisedservos
10−4 theyaretightlycoupled.Insimulationsofclockswithawide
range of atom numbers (i.e. reference signal-to-noise ra-
tios),wefindthattherateatwhichaclockhopstodifferent
10−2 10−1 Ramseyfringesdepends,foragivenLOandafullyoptim-
T/Z isedservo, onlyonthepredictionvariance v. Suboptimal
servos (such as integrators with incorrectly chosen gain)
Figure2. Varianceofservopredictionerrors, expressedasthe
havebothgreaterpredictionvariancevandahigherrateof
variancevofthephaseaccumulatedinaRamseyinterrogation,as
fringehopsforagivenv,sothattheyarelessrobustaswell
afunctionofprobetimeforawhite-noise(top)flicker-(middle)or
asnoisier. Weconjecturethatthebestservosaresimultan-
random-walk-limited(bottom)LO.ThesimulatedclocksuseRam-
eouslythemostrobustandtheleastnoisy,sothatthereis
seyinterrogationof1(solidblackcircles),10(openbluecircles),
102 (solidredtriangles), 103 (opengreentriangles)or104 (solid noneedtochoosebetweenthetwoqualitiesprovidedthat
greysquares)uncorrelatedatoms.Symbolsshowtheperformance onecan,infact,findthisoptimalservodesign.
ofintegratingcontrollersoptimisedwithoutknowledgeoftheLO
noise as discussed in Sec. III. Solid lines show the performance
oflinearpredictorsusingthelastn=50frequencyestimates,de- III. ON-LINESERVOOPTIMISATIONANDNOISE
signedwithknowledgeoftheLOnoisepropertiesasdiscussedin CHARACTERISATION
Sec.II.
In practice, the noise spectrum of the LO may not be
trade-offbetweenaveragingmanymeasurementstoreduce known accurately. A significant benefit of the formalism
theimpactofmeasurementnoiseandconsideringonlythe presentedintheprevioussectionisthatitallowsonetoop-
recentmeasurementsmostrelevanttotheLO’scurrentfre- timisetheservocontrollerwithoutpriorknowledgeofthe
quency. Weknowofnosimple,accurateexpressionforthe
achievableservoperformanceinthisintermediateregime,
buttheroughscalingthatonewouldexpectfromtheafore-
mentionedtrade-off 2 Thisisbestunderstoodbyconsideringtheactionoftheservointhe
frequency domain: the variance of the prediction errors depends on
1 (cid:181)T(cid:182)1+µ/2 thefeedbackgainatafrequencycorrespondingtotheclockcyclerate,
v∝(cid:112) (15)
N Z whereasthesecondintegratoronlycontributesfeedbackgainatmuch
lowerfrequencies.Theeffectofthesecondservoisvisibleinthecorrela-
doesholdinsimulations. tionsbetweenpredictionerrors,notintheirvariance.
6
LOnoiseproperties. Thisispossiblebecausethedefinition 100
ofC inEqn.10involvesonlytheestimatedLOfrequencyer-
T
rorineachclockcycle,whichisroutinelyrecordedinnor- ω
malclockoperation3.Asademonstrationofsuchoptimisa- T)c 10−1
tion,wehaverunclocksimulationswithintegratingservos (L
σ
whosegainswerechosen,withoutknowledgeofthetrueLO d
e
noise,bythefollowingempiricalprocedure: t
uc 10−2
r
1. Start by setting the gain to g =0.2, an arbitrary but st
n
reasonable initial value chosen to allow reliable, if o
c
suboptimal, clock operation under a wide range of Re 10−3
conditions. 10−3 10−2 10−1 100
2. Simulatetheclockfor104 cycles,correspondingtoa TrueσL(Tc)ωT
fewhoursofoperationforatypicalcontemporaryfre-
quencystandard. Figure3. NormalisedAllandeviationofflicker(bluecircles)or
random-walk(redtriangles)LOnoisereconstructedfromcorrela-
3. ComputeC accordingtoEqn.10andthencethevec- tionsintheerrorsignalofasingle-atomclockasafunctionofthe
torw ofoptimalweights.Setg=w 4. actualnoiselevel. Blacklinemarkscorrectreconstruction. Light
1
and dark grey regions mark the noise levels at which the clock
4. Simulate the clock with the newly optimised servo servohopsbetweenRamseyfringesforrandomwalkandflicker
andreoptimise,repeatingasnecessary. noiserespectively.
Evenwhentheservogainisinitiallychosenblindly,wefind
that five rounds of optimisation suffice for the gain g to generallyoptimistic. AsdiscussedinSec.IV,eveninacor-
convergetoavaluethatyieldsperformanceindistinguish- rectly optimised clock there will be unavoidable ambigu-
able from that of an integrator designed with full know- ities in interpreting the error signal (e.g. 2π phase slips
ledgeoftheLOnoisespectrum. Undermorerealisticcon- in Ramsey interrogation) and the resulting measurement
ditions,wheretheinitialchoiceofservoparametersreflects errors contribute to the servo prediction variance without
somepriorknowledgeoftheLOperformance,theoptimisa- beingobservableintheexperimentallyrecordedmeasure-
tioncouldbeperformedmuchmorequickly. Thesymbols mentdata.
inFig.2showthepredictionvarianceofsuchempirically- One can, however, use the correlation matrix C estim-
optimisedintegrators, whichcanbecomparedtotheper- ated during clock operation to partially characterise the
formanceofoptimallinearpredictorsshownassolidlines. LO. Although white noise of the LO is indistinguishable
For clocks operated near their optimal probe times (to be frommeasurementnoiseintheatomicreference,flickeror
discussedinSec.IV),thedifferenceinpredictionvarianceis random-walk noise can produce detectable temporal cor-
lessthan10%.Thus,itispossibletodevelopcontrollersthat relationsevenwhentheircontributionstothetotalmeas-
takefulladvantageofthetimecorrelationsintheLOnoise urementvariancearesmall. ByfittingtheestimatedC toa
evenwithoutindependentknowledgeofthosecorrelations. linearcombination
Unfortunately,itisnotalwayspossibletoverifytheservo
performancedirectly,becausetheobservedvarianceofthe C=σ2(T )C +σ2C +σ2(T )C (18)
w c w f f r c r
error signal e contains contributions both from the servo
predictionerrorandfrommeasurementnoiseoftherefer- of the correlation matrices expected for white, flicker and
ence. Forasingle-atomclockthisproblemisinsurmount- random-walknoise, onecanobtainestimatesoftheAllan
able:theobservedfluctuationsofabinaryerrorsignalmust variancesassociatedwitheachclassofnoiseprocess,which
correspond to quantum projection noise, independent of aresimplythecoefficientsinthelinearcombination.Expli-
servoperformance. Foramany-atomclockwherethede- citexpressionsforthecorrelationmatricesassociatedwith
tectionnoiseiswell-characteriseditispossibletomeasure arbitrarynoisespectraareprovidedinAppendix.A.Fig.3,
the servo prediction variance as an increase in the fluctu- forinstance,showstheflickerandrandom-walkAllandevi-
ations of the error signal, but the resulting estimates are ationsreconstructedinthisfashionfromacorrelationmat-
rixC estimatedfrom2×106 cyclesofoperationofasimu-
latedsingle-atomclock,normalisedtoquantumprojection
noise,asafunctionofthetrueleveloftherespectivenoise
3 Insomeimplementations,theestimatedfrequencymightnotberecor- processes. Random-walk noise, whose correlations differ
dedassuch,butitcanbeobtainedbyaddingtheservopredictionstothe morestronglyfromthoseofthewhitemeasurementnoise,
frequencyerrorsignalreportedbytheatomicreference.
iseasiertodetect,butasseeninFig.3bothflickerandran-
4 Incaseswherethenoiseisnearlywhite,thiscanleadtoatheoretically
dom walk noise can be reliably estimated from levels too
optimalbutpracticallyuselesscontrollerwithnear-zerogain. Toavoid
thisproblem,weimposeaminimumgainof0.04toensureafiniteservo lowtoaffecttheclock’sinstability(variancelessthan1%of
attacktime.Thisboundislowenoughthatitdoesnotaffecttheachiev- projection noise) up to levels that would be unacceptably
ableclockstability. highinnormaloperation,whentheclockservoisjumping
7
1 chain-dotted distribution, corresponding to the distribu-
tion of LO prediction errors. Our best knowledge after a
y measurementonasingleatom,intheeventthatitisfound
t
bili intheexcitedstate,isshownbythedashedprobabilitydis-
a tribution. In this case we can be certain that the phase
b
o was not near φ = −π/2 (since the excitation probability
r
P
would then have been 0), and we can be reasonably con-
n
o fidentthatitliesintheregionbetween0andπ,butwecan-
ti
ta notruleoutthatitliesnear−π,whereboththeexcitation
xci probabilityandthepriorprobabilitydistributionarenon-
E
negligible.Varyingtheprobetime,andhencethespacingof
0 −3π −π −π 0 +π +π +3π theRamseyfringes,involvesatrade-off: increasingT nar-
2 2 2 2
rowsthemainlobeoftheposteriordistribution,thanksto
PhaseErrorφ
thesteeperslopeoftheexcitationsignal,butalsoincreases
the weight of secondary lobes due to other fringes of the
Figure4. Schematicillustrationofinformationgaininasingle
Ramseyspectrum.
cycleofclockoperation.Bluechain-dottedGaussian:Priorknow-
To quantify this trade-off more formally, we consider
ledgeoftheLOfrequency. Blacksolidsinusoid: Ramseyexcita-
Ramseyinterrogationof N uncorrelatedatoms, takingthe
tionspectrum. Reddashedline: PosteriorknowledgeoftheLO
frequencygiventhatanatomwasdetectedintheexcitedstate. If prior distribution (chain-dotted curve in Fig. 4) to be a
noexcitationisdetected,thentheposteriordistributionisthemir- Gaussianofvariancev
rorimageaboutφ=0oftheoneshownhere. −φ2
P(φ)=(cid:112)e 2v . (20)
2πv
between Ramsey fringes. Although this method provides
muchlessdetailedinformationontheLOnoisespectrum Thisdistributionencodes,formally,allthatisknownabout
than does the optical spectrum analyser of Ref. [34], it re- theLOfrequencybeforetheatomicmeasurementresultbe-
quires no measurements beyond those performed as part comesavailable.Inapracticalsense,itisthedistributionof
oftheclock’snormaloperation. Itcanthereforebeusedto servopredictionerrors: iftheservopredictionwereperfect
monitortheLOwhiletheclockisrunning,eveninasingle- (h=x)thenthephaseaccumulatedintheRamseyinterrog-
ionfrequencystandard,providinganearlywarningofper- ationwouldbezero. Theassumptionofnormallydistrib-
formancedegradationsaswellasinformationusefulforthe utedservopredictionerrorsisborneoutbyoursimulations,
optimisationofinterrogationparametersintheatomicref- which show that even with a single-atom reference and a
erence(seeSec.V). binaryerrorsignaltheskewnessandexcesskurtosisofthe
errordistributionarenomorethanafewpercentaslongas
theclockremainslockedtothe(correct)Ramseyfringe.
IV. IMPACTOFSERVOPERFORMANCEONLONG-TERM Thereferencedoesnot,unfortunately,supplyuswiththe
STABILITY expectationvalueP .Rather,themeasurementyieldsaran-
e
domfractionF ofatomsdetectedintheexcitedstatethat
Any measurement performed on a finite number of fluctuatesabouttheexpectationvaluePe duetomeasure-
atomscanyieldonlyafinitenumberofpossibleresults.The mentnoiseoftheatomicreference. Intheabsenceoftech-
optimisationofthemeasurementprotocolthusinvolvesa nicalnoiseonthereferencesignal,thevarianceofF is
compromisebetweenfineresolutionoveranarrowusable (cid:191) P (1−P )(cid:192)
domainofLOfrequenciesandcoarsefrequencyresolution var(F)= (Pe−〈Pe〉)2+ e e (21)
N
overabroaderdomain. Thebestcompromisedependson
e−vsinhv 1−e−vsinhv
therangeoffrequencieswhichmightplausiblyhavetobe = + (22)
measuredbythereference,i.e.onthevarianceofservopre- 4 4N
diction errors. In this section we study this compromise, where the averages 〈·〉 are taken over the prior distribu-
showingthatitleadstoafiniteoptimalprobetimeandan tion P(φ). The first term expresses the fluctuations in the
overalllimitonthelong-termstabilityofclockswithnoisy measured excitation fraction due to actual changes in the
LOs. φ-dependent excitation probability, while the second cor-
Bywayofillustration, considerRamseyinterrogationof responds to quantum projection noise of the binomially-
asingleatom,wheretheprobabilitytofindtheatominthe distributed excitation signal. The usefulness of the excit-
excitedstatedependsonthephaseerrorφ=(x−h)ωT as ationfractioninestimatingthefrequencydependsonthe
1+sinφ covarianceofthetwoquantities
P = . (19)
e 2 cov(φ,F)=(cid:173)φF(cid:174)−〈φ〉〈F〉= ve−v2 (23)
2
ThisexcitationspectrumisshownasasolidlineofFig.4.Let
us assume that, before any measurement, our best know- whichdetermines how muchweight should be given to F
ledgeofthe(corrected)LOfrequencyisrepresentedbythe inconstructingtheposteriorestimateoftheLOfrequency.
8
(cid:48)
Choosingtheweighttominimisethevariancev oftheerror 101 (a)White(µ=−1)
inthisposteriorestimate,wefind(c.f.AppendixC):
v(cid:48)=v−cov(φ,F)2 (24) 100
var(F)
(cid:181) (cid:182)
Nv
=v 1− . (25)
(N−1)sinhv+ev
10−1
This posterior variance combines information from the
measurement with information that was known before-
hand. Inordertoisolatethecontributionoftheformer,we
defineaneffectivemeasurementvariancevmby 101 (b)Flicker(µ=0)
1 1 1
= + . (26)
v(cid:48) v vm τZ 100
(cid:112)
Thisistheusualrelationforthevarianceofa(posterior)es- ω
)
τ
timateobtainedbyanoptimallinearcombinationoftwoin- (
C
dependentpiecesofinformation. vmisthusthevarianceof σ 10−1
ahypotheticalmeasurement,onethatcouldbeinterpreted
withoutanypriorknowledge,andwhichwouldreduceour
uncertaintyontheLOfrequencyasmuchasdidtheactual
measurement.Forthecaseweconsider, 101 (c)Random-Walk(µ=+1)
ev (cid:181) 1(cid:182)
v = + 1− sinhv−v (27)
m
N N 100
1 v2 v3
≈ + + +··· (28)
N 2N 6
where, in the second line, we have expanded the effective 10−1
measurementvarianceinpowersofthepriorvariance.The
firsttermistheconventionalQPNonthemeasurementof
thephase, validwhen v issmallandthecorrectedLOfre- 10−2 10−1 100
quency is known a priori to be well centred on the Ram-
T/Z
seyfringe.Thethirdtermreflectstheadditionaluncertainty
arisingwhenthecorrectedLOfrequencycanlieoutsidethe
Figure5. Long-terminstabilityasafunctionofprobetimefor
range where the reference produces a meaningful result.
clocks using Ramsey interrogation of 1 (solid black circles), 10
Thistermisindependentofatomnumber,anddominates (openbluecircles),102 (solidredtriangles),103 (opengreentri-
theeffectivemeasurementvarianceasvapproaches1. angles)or104(solidgreysquares)uncorrelatedatoms.Solidlines
Replacing the standard phase variance of Eqn. 1 by the showthepredictionofEqn.29,withoutfreeparameters. Vertical
effectivemeasurementvarianceinEqn.2yieldsanewpre- barsmarktherecommendedinterrogationtimegiveninTableII.
dictionforclockstabilityinthelimitoflargeaveragingtime Dashedlinemarksthelimitforperfectphaseestimationwithno
τ,onethataccountsfortheeffectsoflimitedpriorinforma- projectionnoise. Thethreegraphsare,fromtoptobottom,fora
white-,flicker-,orrandom-walk-dominatedLO.
tionineachinterrogation:
(cid:115) (cid:115)
1 T ev (cid:181) 1(cid:182)
σC(τ)→ωT τc N + 1−N sinhv−v. (29) achievablelong-terminstabilityasafunctionofthechoice
ofprobetime.
In Fig. 5 we compare this prediction with simulations for Whentheprobetimeisshort,theRamsey(cid:112)fringeisbroad
clockswithwhite-,flicker-,andrandom-walk-noise-limited andv issmall,theinstabilityimprovesas1/ T asconven-
LOs, using Ramsey interrogation of uncorrelated atoms tionallyexpected. Theimprovementwithincreasingprobe
withnodeadtime(T =T).Eachpointinthesegraphsisob- time stops either when the additional v-dependent terms
c
tainedfromasimulationof2×106cyclesofclockoperation. inEqn.29growimportantorwhentheservocannolonger
TheAllandeviationiscomputedforatimeτlon(cid:112)genough reliablylocktheLOtothereferencetransition: thecurves
thattheinstabilityhasreachedtheasymptotic1/ τregime inFig.5endwhenthefringe-hopratereaches1per2mil-
(corresponding to 2×104 cycles of clock operation), then lioncycles.AsN increases,quantumprojectionnoiseisre-
rescaled to a fixed averaging time Z and normalised to a ducedrelativetoLOnoiseanditbecomesadvantageousto
fixednoiselevelσ (Z)toobtainadimensionlessresultthat reducetheprobetimesoastobelesssensitivetothelatter.
L
is comparable across systems. The graphs thus show the Thus, the optimal probe time gets shorter with increasing
9
LONoiseType Asymptoti(cid:112)cScaling LONoise Safe AsymptoticOptimum
ofσC(τ)ω Zτ T/Z T/Z
White ∝N−1/3 White – min(cid:179)N−1/5,1.4N−1/3(cid:180)
Flicker ∝N−5/12 Flicker 0.4−0.15N−1/3 0.76N−1/6
Rnd.Walk ∝N−4/9 Rnd.Walk 0.4−0.25N−1/3 0.79N−1/9
Table I. Asymptotic scaling of LO-limited clock instability with TableII.RecommendationsforthechoiceofRamseyinterrogation
atom number. The scaling differs from the conventional N−1/2 time. Thelastcolumngivestheoptimalprobetimeinthelimitof
QPN limit because the optimum probe time decreases with in- manyatoms.Thereisnothingtobegainedbyprobinglongerthan
creasingatomnumber. thistime. Itmaybenecessarytouseshorterprobetimestoavoid
fringehops; asuggestedsafeupperboundontheprobetimeis
giveninthesecondcolumn.
atomnumber,andthefullyoptimisedclockinstabilitydoes
notscaleasN−1/2. TheasymptoticscalingwithN isgiven
inTableI.Forwhitenoise,themostextremecase,theop- obtaintheasymptoticallyoptimalprobetime(lastcolumn
timalprobetimescalesasN−1/3 inthelarge-N limit,lead- ofTableII).Increasingtheprobetimebeyondthisoptimum
ingtoaN−1/3scalingofthelong-termAllandeviation. For alwaysleadstoanincreaseineffectivemeasurementvari-
flickerorrandom-walknoise,v fallsoffmoresteeplyasthe ance and long-term instability, and is of no practical in-
probe time is shortened (see Eqn. 13), so that the the op- terest. At small atom numbers, shorter probe times are
timalprobetimeislesssensitivetoatomnumberandascal- requiredtokeeptheservocontrollerrobustagainstfringe
ingclosertotheconventionalQPNlimitisobtained. Inthe hops. Thepurelyphenomenologicalboundinthesecond
absence of projection noise, i.e. in the limit N →∞, the columnofTableIIischosentobeslightlyshorterthanthe
servoperformancelimitofEqn.13combineswithEqn.29 time for which we observe fringe-hops at a rate of 1 per
toyieldageneralmeasurement-noise-independentlimiton million simulated clock cycles, with a 20% safety margin.
clockinstability,whichweplotasdashedlinesinFig.5.This Ourchoiceofmaximumacceptablefringe-hoprate,corres-
limitarisessolelyfromtheunpredictabilityoftheLOnoise ponding to a requirement that the clock remain locked to
andfromthefinitedomainoverwhichtheRamseyerrorsig- thecorrectfringeforafewdays,isarbitrary,butastheonset
nalcanbeunambiguouslyinterpreted. offringe-hoppingisextremelysteep(thefringe-hopratein
Strictlyspeaking,Eqn.24holdsonlyiftheestimatedfre- simulationsincreasesbytwotothreeordersofmagnitude
quency error is a linear function of the measured excita- whentheprobetimeisdoubled),themaximumsafeprobe
tionfraction.Sincetheexcitationprobabilityisanon-linear timeisonlyweaklydependentonthischoiceofthreshold.A
(e.g. sinusoidal) function of the LO frequency error, one fulloptimisationofallcommonprobeprotocolsinthepres-
might hope to do better than the estimated performance enceofrealisticexperimentalimperfectionsisbeyondthe
of Eqn. 29 by using a non-linear function to convert the scopeofthiswork,butweexpectqualitativelysimilarbeha-
excitation fraction to a frequency error estimate. Simula- viourfromRabiorhyper-Ramseyprobing,withsomewhat
tions show, however, that correcting for the curvature of longeroptimalprobetimesandslightlydegradedinstability
theRamseyfringebyestimatingtheaccumulatedphaseas due to the increased width of the observed atomic reson-
arcsin(2F−1)ratherthansimply2F−1hasnosignificantef- anceinthesesschemes. Conversely,weexpectthatclocks
fectonvorontheachievablelong-termclockstability.One withsignificantdeadtimeintheiroperatingcyclewillneed
canunderstandthisfindingbynotingthat,whenv islarge tousesomewhatshorterprobetimestocompensateforthe
enoughthatthecurvaturewithinasingleRamseyfringeis servo’sinabilitytocorrectunobservedLOfrequencyfluctu-
significant, the effect of unavoidable ambiguities such as ations,whichwillleadtoav higherthaninourdead-time-
thesecondarylobeinFig.4ismuchlargeranddominates freesimulations.
theposteriorvariance.
VI. CONSTRAINTSONTHEBENEFITSOFENTANGLED
V. GUIDELINESFORINTERROGATIONPARAMETERS ATOMICREFERENCES
To choose the operating parameters for a clock, one Theargumentsdevelopedintheprecedingsectionsalso
can in general use the formalism of Sec. II to predict the apply to certain Ramsey-like protocols using entangled
servo error variance v as a function of those operating states in the atomic reference, provided that LO noise is
parametersandEqn.29topredicttheresultinglong-term thelimitingformofdecoherence.Forinstance,thescheme
stability, which can then be optimised. Table II, for ex- proposed in Ref. [35] and demonstrated in Refs. [36–38],
ample,providesrecommendedRamseyinterrogationtimes employing N ato(cid:112)msinamaximallycorrelatedstate[|ψ〉=
for clocks dominated by different types of power-law LO (cid:161)|g〉⊗N+|e〉⊗N(cid:162)/ 2, where |g〉 and |e〉 are the atomic ei-
noise, expressed as multiples of Z. In the many-atom genstates], isfullyequivalenttoRamseyinterrogationofa
limit, the servo prediction errors become independent of singleatomwithanN-foldenhancedtransitionfrequency
thequantumprojectionnoiseandwecansolveEqn.29to by the corresponding harmonic of the LO radiation. Now
10
thelong-terminstabilityofasingle-atomfrequencystand- 102
ardcanbeexpressedas (a)White(µ=−1)
s 101
σ (τ)= (cid:112) , (30)
C ω Zτ 100 τ1
withsadimensionlessconstantoforderunityencodingthe
choiceofprobetimeT/Z andtheadditionalcontribution 10−1 τ
2
ofLOnoiseatthisprobetime. Thelong-terminstabilityof
theclockusingmaximally-correlatedatomsthusbecomes 10−2
σ (τ)= (cid:112)s , (31) 10−3
C Nω ZNτ 102
(b)Flicker(µ=0)
where Z isthenoisetimescaleforthe N-foldfrequency-
N
multipliedLO: 101 τ1
σL(ZNc)NωZN=1rad. (32) ωZ 100 τ
τ) 2
In the dead-time-free limit ZNc = ZN, one can compare (C 10−1
σ
Eqn.32withEqn.3andfind
10−2
ZN =N2−+2µ, (33)
Z 10−3
withµagaindescribingthetime-dependenceoftheLOAl- 102
lanvariance(seeSec.I).Theentangledclock’slong-termin- (c)Random-Walk(µ=+1)
stabilitythusscalesas 101 τ
1
s 1
σ (τ)= (cid:112) . (34) 100 τ
C ω ZτN12++µµ 2
10−1
Thus,iftheclockstabilityislimitedbywhiteLOnoise(µ=
−1), a reference using a maximally-correlated state of N 10−2
atoms performs no better than a reference using a single
atom,andisinfactworsethanareferenceusinguncorrel- 10−3
atedinterrogationoftheNatoms.Thisscalinghasbeenob- 10−4 10−2 100 102 104
servedexperimentallyforcorrelatedmagneticfieldnoisein τ/Z
a14ionGHZstate[38].Forflicker-floorLOnoise,theAllan
deviation improves as N−1/2 with N maximally-entangled Figure6. SimulatedAllandeviationasafunctionofaveragingtime
atoms,veryslightlybetterthantheasymptoticN−5/12scal- for optimised clocks using either 100 uncorrelated atoms (solid
ing achievable without entanglement, but worse than the bluecircles)oramaximally-correlatedstateof100atoms(open
scalingachievedwithunentangledatomsforN <102. Itis red circles). The latter must operate with shorter interrogation
only for random-walk LO noise that maximally entangled times,andthusthecorrespondingcurvesstartearlier. Theblack
states offer measurable benefits, with an N−2/3 scaling of linemarkstheAllandeviationofthefree-runningLO.Thethree
graphsare,fromtoptobottom,forawhite-,flicker-,orrandom-
thelong-termAllandeviation. Weillustratethesescalings
walk-dominated LO. Arrows mark the two integrator time con-
inFig.6,whichplotstheAllandeviationspectrumrecorded
stantsforthecaseofuncorrelatedatoms,asdiscussedinSec.VII.
insimulationsoffullyoptimisedclocksusingeither100un-
correlatedatomsora100-atommaximally-correlatedstate
forallthreeLOnoisetypes.
Maximallyentangledstatesreducethesignal-to-noisera-
tioofameasurementonN atomstothatofasinglequbit,
providing less new information per measurement but ac- as in Ramsey interrogation of uncorrelated atoms, and so
celeratingtheclockcyclesothatmoremeasurementscan theyaresubjecttothesameprojection-noise-independent
be averaged. That is why their use is advantageous with sinhv−v≈v3/6limitontheireffectivemeasurementvari-
random-walkLOnoise, whenfastmeasurementscantake ance(c.f. Eqn.28). Theadditionalnoiseintroducedwhen
advantage of the reduced LO noise at short time scales. servopredictionerrorsallowtheanti-squeezedquadrature
Otherapproachestotheuseofentanglementinatomicref- tocontaminatethemeasurementresult[42]caninprinciple
erences, such as spin squeezing [24–26, 39–41], focus in- beeliminatedbyasuitablereadoutprocedure[27]inwhich
stead on improving the signal-to-noise ratio of the meas- caseweexpectsuchinterrogationprotocolstoofferbenefits
urement, and thus increasing the amount of new inform- comparabletosuppressingtheprojectionnoisebyincreas-
ationobtainedineachinterrogation. Theerrorsignalpro- ingatomnumber,evenwithwhite-orflicker-floorlimited
ducedinsuchschemeshasthesameperiodicambiguities LOs.