A strontium optical lattice clock with 1× 10 −17 uncertainty and measurement of its absolute frequency
Richard Hobson, William Bowden, Alissa Silva, Charles F. A. Baynham, Helen S. Margolis, Patrick E. G. Baird, Patrick Gill, Ian R. Hill
AA strontium optical lattice clock with × − uncertainty and measurement of its absolutefrequency Richard Hobson , ∗ , William Bowden , , ∗ , Alissa Silva ,Charles F. A. Baynham , , Helen S. Margolis , ,Patrick E. G. Baird , Patrick Gill , , Ian R. Hill National Physical Laboratory, Hampton Road, Teddington TW11 0LW, UnitedKingdom Clarendon Laboratory, Parks Road, Oxford OX1 3PU, United Kingdom ∗ Both authors contributed equally to this work.
Abstract.
We present a measurement of the absolute frequency of the 5s S to 5s5p P transition in Sr, which is a secondary representation of the SI second.We describe the optical lattice clock apparatus used for the measurement, andwe focus in detail on how its systematic frequency shifts are evaluated witha total fractional uncertainty of 1 × − . Traceability to the InternationalSystem of Units is provided via comparison to International Atomic Time (TAI).Gathering data over 5- and 15-day periods, with the lattice clock operatingon average 74% of the time, we measure the frequency of the transition tobe 429 228 004 229 873 . × − . We describe in detail how this uncertainty arises from theintermediate steps linking the optical frequency standard, through our local timescale UTC(NPL), to an ensemble of primary and secondary frequency standardswhich steer TAI. The calculated absolute frequency of the transition is in goodagreement with recent measurements carried out in other laboratories around theworld. a r X i v : . [ phy s i c s . a t o m - ph ] M a y strontium optical lattice clock with × − uncertainty and measurement of its absolute frequency
1. Introduction
The precision of optical atomic clocks based on trappedions [1, 2] and neutral atoms [3–5] is making rapidprogress, with estimated systematic uncertainties nowroutinely lower than those of the most accurate Csfountains [6–8]. As well as potentially supportingtime scales [9–12] and a future redefinition of theSI second [13, 14], optical clocks are also promisingtools for geodesy [15, 16] and for carrying out tests offundamental physics including violations of relativity[17–19], signatures of dark matter [20, 21], or timevariation of fundamental constants [22, 23].One of the most successful approaches to atomicclock-making has been to use an optical lattice clock(OLC) configuration, in which a highly forbidden S to P transition is probed in an ensemble of laser-cooled neutral atoms trapped in a magic-wavelengthoptical lattice [24]. OLCs are being pursued basedon a wide range of alkaline earth-like atomic species,including Yb [25–28], Hg [29,30], Cd [31], and Mg [32].In this report we present results from an OLC based on Sr, contributing a new data point to a rich history ofabsolute [33–47] and relative [4,15,29,48,49] frequencymeasurements with the same atom.This report is organized as follows. In section 2 weoutline the optical lattice clock apparatus. In section 3we describe how the systematic frequency shifts andtheir uncertainties are evaluated. Finally, in section 4we describe the absolute frequency measurement of theoptical lattice clock against International Atomic Time(TAI).
2. Experimental apparatus
A detailed overview of NPL’s Sr optical lattice clockhardware has been given in previous work [50, 51].Only some key parts of the experimental apparatusare recapitulated here.The Sr atomic source consists of an oven heated toapproximately 800 K, followed by a 30 cm transverse-field permanent-magnet Zeeman slower, details ofwhich have previously been presented in [52]. Theatomic beam then enters the science chamber, a steelspherical octagon with radius 112 mm, in which theatoms are captured and then cooled in a two-stagemagneto-optical trap (MOT). The first ‘blue’ MOToperates for between 100 ms and 300 ms on the 5s S to 5s5p P transition at 461 nm. The large scatter rate of 2 π ×
30 MHz on this blue transition enablesthe efficient capture of atoms from the slowed atomicbeam, but limits the MOT temperature to around2 mK. In order to prevent atoms from being shelvedfrom the blue MOT into the 5s5p P , states, repumplasers are applied at 497 nm and 679 nm, enhancingthe blue MOT lifetime from around 18 ms to 2 s.For efficient repumping of Sr, transitions out of allfive hyperfine P states are addressed by tuning the497 nm laser to the F = 11 / F (cid:48) = 11 / . Sr, can befound in Ref. [53].After the blue MOT, a ‘red’ MOT is operated fora total of 230 ms on the 5s S to 5s5p P transitionat 689 nm. The low scatter rate of 2 π × . µ K rangein a two-stage sequence [54, 55]. An initial broadbandred MOT stage lasting 80 ms, where the cooling laseris frequency modulated to cover a 2 MHz spectrum,captures hot atoms from the blue MOT. Next, asingle-frequency red MOT lasting 160 ms further coolsthe atoms to 2 µ K and compresses the cloud to adiameter of around 200 µ m, at which point the atomsautomatically load into the co-located optical latticetrap. Throughout the red MOT, two separate laserfrequencies are applied so that the F = 9 / F (cid:48) = 9 / F = 9 / F (cid:48) = 11 / µ m at the atomsbefore being collimated and retro-reflected to form a1D standing wave. We use a Ti:Sapphire laser asthe 813 nm source, which is frequency stabilized to atransfer cavity whose length is itself stabilized to the698 nm clock laser to ensure long-term stability. Beforethe beam is delivered to the atoms, it propagatesthrough a volume holographic grating optical bandpassfilter with 12 GHz full width half maximum, whichstrongly suppresses any spectral impurities arisingfrom laser-amplified spontaneous emission or anyother sources. The system can supply up to 1 Wat the atoms, generating lattice trap depths up to U = 200 E r , where E r = h × . strontium optical lattice clock with × − uncertainty and measurement of its absolute frequency S M F = 9 / S M F = − / σ + or σ − polarized light at 689 nm addressing the5s S F = 9 / P F (cid:48) = 9 / µ K: thelattice depth is linearly ramped down to 26 E r in 20 ms,held for 20 ms to let the hot atoms escape, and thenramped back up to the operating depth of 52 E r forclock spectroscopy. Third, a 22 ms Rabi π pulse isimplemented at 698 nm in a quantization field of 72 µ T,resonant with the 5s S M F = ± / P M (cid:48) F = ± / M (cid:48) F = 9 / M (cid:48) F = − / π pulseusing a 698 nm clock laser, and the resulting excitationfraction is used to steer the laser toward the atomicresonance. The clock laser consists of a home-builtextended-cavity diode laser (ECDL) which is pre-stabilized with 1 . × − . The laser phases are comparedover a self-referenced optical frequency comb in asimilar arrangement as in Ref [58], exploiting a transferoscillator scheme [59] to cancel noise in the combrepetition rate.
3. Systematic frequency shifts
There are several systematic frequency shifts influenc-ing optical lattice clocks, all of which must be charac-terized and corrected for in order to realize an accuratefrequency standard. Here we describe how these shiftsare characterized in our lattice clock system, reachinga total systematic uncertainty of 1 . × − . The in-dividual contributions to the uncertainty budget areoutlined in table 1. † The BBR correction and uncertainty change with time—seetext ‡ The servo error is below 2 × − for datasets longer than1 hour—see text Table 1.
Uncertainty budget for the NPL Sr lattice clock.Reported uncertainties correspond to 68% confidence intervals.All values are in units of 1 × − . Systematic effect Correction UncertaintyBBR chamber † ‡ Total Correction
The largest systematic shift in the Sr optical latticeclock is from the blackbody radiation (BBR) emittedby the room-temperature vacuum chamber. FollowingRef [60], the BBR-induced fractional frequency shiftcan be split into static and dynamic components: y BBR = β st (cid:18) TT (cid:19) (cid:34) η (cid:18) TT (cid:19) + η (cid:18) TT (cid:19) + η (cid:18) TT (cid:19) (cid:35) (1)where we have chosen an arbitrary reference temper-ature T = 300 K. Higher-order dynamic correctionsare below 1 × − , while magnetic dipole and elec-tric quadrupole interactions with the BBR field con-tribute only at the 6 × − level and can thereforebe neglected [61].The static coefficient in Equation 1 can becalculated from the DC polarizability measurementin Ref [62] as β st = − . × − . Thedynamic corrections can be calculated from various linestrength data, with the largest contribution being fromthe 2 . µ m 5s5p P to 5s4d D transition measuredin Ref [63], giving β st η = − . × − , β st η = − . × − and β st η = − . × − . (Note that the uncertainties in these dynamiccoefficients are strongly correlated, so we calculatea total uncertainty by summing linearly rather thanin quadrature). At our operating temperature of294 K, the total systematic uncertainty introduced dueto imperfect knowledge of the BBR coefficients is1 . × − .However, the main limitation in our lattice clocksystem is not from theory, but rather from theexperimental uncertainty in the BBR temperature T /T . To measure this temperature, eleven Pt100 strontium optical lattice clock with × − uncertainty and measurement of its absolute frequency T max and the lowest T min .This is in accordance with BIPMs GUM: Guide tothe Expression of Uncertainty in Measurement whichrecommends adopting such a distribution when onlythe bounds on a quantity are known [65]. This yieldsan estimated temperature of ( T max + T min ) / T max − T min ) / √ × − .As a consistency check we also developed analternative model for the total BBR field using a solid-angle- and emissivity-weighted sum of contributionsfrom the various components of the science chamber,similar to the approach presented in Ref [66]. In atypical experimental run the two models agree witheach other to within much less than their uncertainties.However, we conservatively base our final estimate onthe rectangular distribution model since it returns alarger uncertainty and doesn’t rely on assumptionsregarding material properties and geometries unlike theemissivity-weighted, solid-angle approach.Finally we must consider the contribution to theBBR environment from the Sr oven source, whichis heated to around 800 K and has a direct line ofsight to the atoms. We model this effect using asimilar approach as for the leakage into the cryogenicBBR enclosure in Ref [67], but in our system wefind a negligible oven contribution on the order of2 × − —the BBR leakage into the main chamberis strongly suppressed by the presence of two 1 mm-diameter apertures along the atomic beam, both placed Figure 1.
The lower plot shows the fractional frequency BBRcorrection over the first 10 days of the campaign along with thecorresponding estimate of the equivalent operating temperature.The blue shaded region represents the 1-sigma uncertainty whichis also plotted in the upper plot for clarity. This uncertainty isprimarily set by the maximum gradient across the chamber. before the Zeeman slower at a distance of around 0 . . The clock transition is interrogated in a bias field of72 µ T in order to ensure that the different Zeemantransitions are well resolved from each other. Withthe bias field applied, the M F = 9 / M (cid:48) F = 9 / . × − µ T − in fractional frequency units [68, 69].In order to provide immunity from the linearZeeman shift, two clock servos are combined with spin-polarized samples of atoms in the M F = +9 / M F = − / ν avg = (cid:0) ν +9 / + ν − / (cid:1) /
2, for which the linear Zeemanshift cancels out. In theory, this cancellation couldbe compromised by drift in the bias magnetic fieldbetween consecutive servo cycles; however, we observea sufficiently small magnetic field drift of 20 nT over10 s to imply a negligible effect from drifts in the linearZeeman shift.By contrast, the quadratic Zeeman shift is notcancelled by averaging the two stretched transitions.To characterize the quadratic shift, the frequencysplitting ν +9 / − ν − / is continuously logged andthe shift coefficient of − . × − kHz − [63] isapplied. For the splitting of 706 Hz used during themeasurement in this report, we calculate a systematicshift of − . × − . strontium optical lattice clock with × − uncertainty and measurement of its absolute frequency The optical lattice trap is operated close to the magicwavelength where the shifts of the excited and groundstates should be the same [24]. Nonetheless, there areseveral systematic shifts caused by the lattice trappingfield, for example: residual electric dipole (E1) shiftsdue to deviations from the magic wavelength; magneticdipole (M1) and electric quadrupole (E2) interactions;and fourth-order perturbations from the electric dipoleinteraction (also known as hyperpolarizability). Sincethe different interaction terms have different spatialprofiles in the 1D lattice, a complete treatment musttake into account details of the atomic motion withinthe lattice potential [71].In this report we apply a simplified model ofthe lattice shift, neglecting some of the higher-ordermotional corrections which only contribute below the2 × − level in our operating conditions. In thesimplified model, the total fractional frequency shiftcan be written: y L = ∆ α (cid:48) E1 (cid:18) U eff E r (cid:19) + β (cid:48) (cid:18) U eff E r (cid:19) + ∆ α (cid:48) E2M1 (cid:18) n z + 12 (cid:19) (cid:18) U E r (cid:19) (2)where n z is the axial motional state, and we definean effective trap depth U eff = U − kT r to takeinto account that the finite radial temperature T r reduces the mean lattice intensity seen by the atoms—in a classical approximation, using the equipartitiontheorem, the atoms possess a potential energy of kT r / T z ≈ . µ K is not included whencalculating the effective trap depth since along thelattice axis the atoms are in the non-classical regime (cid:104) n z (cid:105) ≈ . (cid:28) α (cid:48) E2M1 = 0 . × − and hyperpolariz-ability coefficient β (cid:48) = 1 . × − , as measuredin Ref. [72]. The residual E1 coefficient ∆ α (cid:48) E1 , whichdepends strongly on local operating conditions such asthe lattice wavelength, is extracted by fitting to fre-quency shift data between interleaved servos at differ-ent lattice depths ranging between 52 E r and 156 E r .The frequency instability of one of these interleaveddatasets is shown in figure 2. The fitting script is runmultiple times in a Monte Carlo simulation using ran-domly generated E2M1 and hyperpolarizability coeffi-cients, so the uncertainties in these higher-order shiftsare propagated appropriately. Motional sideband scansare implemented at each operating depth in order toevaluate U , T z and T r [73], and the lattice wavelength Averaging Time / s O v e r l a pp i n g A ll a n D e v i a t i o n ( ) = 1.6 × 10 / Detuning / Hz Figure 2.
Overlapping Allan deviation of the frequencydifference between two interleaved atomic servos operating atdifferent lattice depths over a continuous 14 h period. Bymeasuring the offsets between the servos we can estimate thelattice induced Stark shift under normal operating conditions.For our system the interleaved instability, shown in red, is1 . × − / √ τ . Inset shows the line shape for a 300 ms Rabipulse which has a Fourier limited linewidth of 2 . and lattice polarization (parallel with the magneticfield) are both kept constant throughout all measure-ments. Combining several days of interleaved latticeshift data, we reach a total shift of − . × − at the operating depth of 52 E r .As noted in Ref [63], the lattice shift measurementcan be distorted by parasitic collisional effects: adeeper lattice will compress the atoms into a smallervolume, causing a differential collisional shift thatcould be mistaken as a lattice shift. Furthermore,loading atoms into a deeper trap or ramping thetrap depth after loading can change the atomictemperature—in turn changing the collisional shift[71]. To estimate how this affects the measuredlattice shift, we run the Monte Carlo simulation usingtwo different models for the scaling of atom density ρ ( U ) with trap depth: ρ ∝ U / , which assumes T z and T r are constant with trap depth, and ρ ∝ U / , which assumes the mean thermal occupancynumber is constant. The latter closely resembles ourexperimental conditions, since we load atoms intothe lattice at a fixed trap depth before adiabaticallyramping to the final set point used during spectroscopy.Based on data from our independent collisional shiftevaluation in section 3.4, and by operating with areduced atom number at higher lattice set points,we estimate that the uncertain relationship betweenthe collisional shift and the trap depth introduces anuncertainty in the estimated lattice shift of 1 × − . strontium optical lattice clock with × − uncertainty and measurement of its absolute frequency The cooling sequence described in section2 prepares high purity samples of atoms in an identicalinternal state, implying that s -wave collisions should bestrongly suppressed by the Fermi exclusion principle.Nonetheless we search for any small residual s -waveor p -wave collisional shifts by measuring the frequencydifference between interleaved servos containing atomnumbers N and 4 . N respectively. The observedfrequency difference of − × − between theseservos implies a systematic shift at our operating atomnumber N of − . × − . Background gas collisions:
In our science chamberthe vacuum-limited lifetime both for magnetically-trapped 5s5p P atoms and for lattice-trapped 5s S atoms is observed to be 8 s. Applying the modelin Ref. [74], and assuming a background dominatedby hydrogen (or by other gases with similar C coefficients [75]), the associated collisional frequencyshift is estimated as − × − . Since this model ofthe background gas shift is yet to be experimentallyverified, the uncertainty in the shift is also taken to be2 × − . DC Stark shift:
We evaluate the backgroundelectric field using spectroscopy on a Rydbergtransition to 5s75d D , as previously described in[76]. The large steel chamber with comparatively smallviewports proves to be quite an effective shield againstbackground electric fields, yielding a systematic shiftof − . × − . Probe Stark shift:
We estimate the probe beamintensity required to drive the 200 ms Rabi π pulse onthe clock transition using the experimentally measurednatural lifetime of the 5s5p P state of 330(140) s [77].We also include the Clebsch-Gordan coefficient 0.9045and the Lamb-Dicke parameter 1 − η z = 0 .
90, bothof which slightly increase the required intensity. Thisestimated intensity, when multiplied by the theoreticaldifferential polarisability at 698 nm between the twoclock states [78], results in an estimated probe-inducedStark shift of − . × − . Doppler shifts:
To a good approximation, theposition of each atom is defined by the position of thelattice site in which it is trapped. Since the latticewavelength is well controlled, the motion of the latticesites is mostly set by the motion of the lattice retro-reflecting mirror. To avoid first-order Doppler shifts,it is therefore important to make sure that the clockprobe beam phase is as stable as possible relative tothe lattice retro-reflector. In our system, we activelyphase-stabilize the delivered clock probe light [79] ata reference sampler placed close to the lattice retro-reflector, leaving only a short uncompensated pathof 20 cm in free space. The beat signal used inthis phase stabilization loop is logged on a frequency counter throughout any measurement so that any cycleslips can be detected and the corresponding datapoints discarded (though normally no such glitchesare detected). The second-order Doppler shift fromthermal motion at the atomic temperature of 2 µ K isbelow 10 − and is therefore also omitted from table 1. AOM phase chirp:
In order to suppress theheating-induced phase chirp from the final ‘switching’acousto-optic modulator (AOM), the reference samplerfor phase stabilization is placed after the AOM, retro-reflecting the 0 th -order beam. The idea behind thisarrangement is that the 0 th -order beam traverses avery similar path through the AOM as the 1 st -orderbeam which probes the atoms. The effectiveness of thiscompensation technique has been verified by turningoff the phase stabilization servo and measuring thephase chirp during the probe pulse at high RF drivepower against a separately delivered phase-stabilizedbeam. With the servo re-engaged, we observe thatthe phase stabilization loop suppresses the AOM phasechirp by at least a factor of 5 (with this factorlimited by statistics in the measurement). Combinedwith the reduced RF drive power used during clockoperation, which results in an experimentally verifiedproportionate reduction in the phase chirp, we estimatea systematic shift from AOM phase chirp of less than3 × − . Line pulling:
The nearest π -polarized transitionis M F = 7 / M (cid:48) F = 7 /
2, which is split from themain clock transition by 67 Hz. When scanning theclock laser over the expected M F = 7 / × − . If the probe beam were notperfectly π -polarized, then it would also be possibleto observe line-pulling effects from the M F = 9 / M F = 7 / − . Servo Error:
The clock servo acts to ensurethat the local oscillator is steered exactly to atomicresonance, but lock offsets can remain due to finiteservo gain which allow the local oscillator to sagabove or below the clock transition frequency. Tocharacterize the error due to finite servo gain, theatomic excitation is recorded and post-processed torecalculate the error signal and therefore the meanfrequency offset. We observe that on average the servoerror scales as the inverse of the measurement time,and for datasets longer than 1 h the servo offset isconsistently below 2 × − . strontium optical lattice clock with × − uncertainty and measurement of its absolute frequency
4. Absolute frequency measurement
Determination of the absolute frequency of thestrontium optical lattice clock requires measurementagainst a realization of the SI second. Thiscan be achieved either via direct comparison to alocal caesium primary standard, or indirectly bymeasuring the frequency versus International AtomicTime (TAI), which also provides traceability to theSI second. Historically, the majority of absolutefrequency measurements have followed the directapproach, but more recently several groups reportedvalues which were derived via comparison to TAI[27,80–82], in some cases attaining uncertainties below5 × − [44, 83] or even surpassing the best directmeasurements using local primary frequency standards[84]. For the measurement in this report we adoptedthe latter approach, accessing the SI second throughcomparison against TAI.The absolute frequency measurement is performedin several stages. First, the optical frequency must becompared to the scale interval of our local time scaleUTC(NPL). Secondly, the local time scale is comparedagainst TAI. This latter comparison is complicatedby the fact that TAI is a virtual time scale that iscomputed on a monthly basis by the InternationalBureau of Weights and Measures (BIPM), as anintermediate step in the computation of CoordinatedUniversal Time (UTC). The time offset of UTC(NPL)from UTC is published at 5-day intervals in section 1 ofBIPM’s monthly Circular T bulletin. This means thatit is only possible to calculate the frequency offset of thelocal time scale from TAI averaged over 5-day intervals.(Note that the scale interval of UTC is identical tothat of TAI, since the two differ only by an integernumber of leap seconds.) Finally, a correction mustbe applied to account for the deviation d of the scaleinterval of TAI from the SI second, published only asa monthly average in section 3 of Circular T. If theperiod of an optical frequency measurement does notcoincide with these monthly reporting periods, it isalso possible to request a custom computation of d for specific periods corresponding to one or more 5-day Circular T reporting intervals [85]. The procedurelinking the local optical frequency reference to the SIsecond can be summarized by the following expression: f Sr f SI = f Sr f UTC(NPL) × f UTC(NPL) f TAI × f TAI f SI (3)where f (SI) = 1 Hz by definition. In practice,the combination of optical clock downtime and thefrequency instability of the local time scale complicatesthe measurement. This downtime can give rise tomeasurement error as the mean frequency of the localtime scale during the time period when the optical clock was operational ( T UP ) may differ from its averageover the total measurement period ( T ALL ). To accountfor this effect, we modify equation 3 as follows: f Sr f SI = f Sr f UTC(NPL);T UP × f UTC(NPL);T UP f UTC(NPL);T
ALL × f UTC(NPL);T
ALL f TAI;T
ALL × f TAI;T
ALL f SI . (4)Using equation 4, we determine the absolute frequencyof the 5s S to 5s5p P transition in Sr overtwo separate periods during the month of June 2017(MJD 57904-57918 and MJD 57929-57933) as shownin figure 3. Below we describe how a value for eachratio in the above expression is determined for thesetwo measurement intervals. f Sr /f UTC(NPL);T UP The local frequency ratio f Sr /f UTC(NPL);T UP is deter-mined by using a femtosecond optical frequency combreferenced to a 10 MHz signal produced by a hydrogenmaser. The same maser is used to generate a one pulse-per-second signal from its 10 MHz output which servesas the basis of UTC(NPL), hereby providing a directlink between the optical frequency and the local timescale. The frequency comb has been verified to intro-duce negligible uncertainty in such optical-microwavefrequency comparisons. The RF beat signals are π -counted, with a one second gate time, using K+K FXEfrequency counters.The main source of uncertainty in this measure-ment comes from the distribution of the 10 MHz masersignal to the frequency comb laboratory, and the sub-sequent synthesis in that laboratory of an 8 GHz sig-nal against which the repetition rate of the frequencycomb is measured. Potential time-varying phase shiftsare monitored by dividing the 8 GHz signal by 800and comparing the resulting 10 MHz signal with theoriginal signal from the hydrogen maser using a phasecomparator. Based on this round-trip data, we esti-mate the RF distribution and synthesis to contributean uncertainty of 1 × − to the frequency ratio mea-surement.We also account for uncertainty introduced by thefrequency instability of the optical frequency standard.Extrapolating the estimated white frequency noise as2 × − τ − / , a conservative upper bound based onmeasurements against another OLC over an opticalfibre network [17], for the total measurement uptime,we estimate the statistical uncertainty arising fromfrequency instability of the OLC to be 2 × − and3 . × − for the two measurement periods. strontium optical lattice clock with × − uncertainty and measurement of its absolute frequency MJD - 57904 -2x10 -1x10 f S r / f U T C ( N P L ) ; T U P (a) Averaging Time (s) O A D E V o f t h e r a t i o f S r / f U T C ( N P L ) ; T U P (b) Noise Model Sr vs Maser Sr vs Maser (Max DT: 2500s) N u m b e r o f O cc u rr e n c e s
74% Uptime (c) UptimeDowntime
Figure 3.
Figure (a) shows the ratio between the frequency of the optical clock transition in Sr and the frequency of the hydrogenmaser as measured over the two campaign periods. The red curve represents a moving average over a 100 second window. Figure (b)shows the fractional frequency instability of the ratio between the OLC and the hydrogen maser averaged over the entire campaignas measured by the overlapping Allan deviation. The red Allan deviation points are based only on continuous datasets while greenpoints are based on quasi-continuous datasets which can have downtime periods of less than 2500 seconds (see text for more details).From this instability data, we generate the stochastic noise model (blue curve) used in the Monte Carlo modelling of the uncertaintyarising from downtime in the measurement. (c) Distribution of all continuous periods of uptime and downtime exceeding 10 minutesin length. The majority of downtime results from the unlocking of one of the lasers needed for cooling and trapping the Sr atoms.For most of these lasers, we have automatic recovery systems that can reacquire lock quickly without human intervention. As aresult of this automation, the median downtime period over the entire campaign is 20 seconds. f UTC(NPL);T UP /f UTC(NPL);T
ALL
To estimate the correction and uncertainty associatedwith the ratio of f UTC(NPL);T UP to f UTC(NPL);T
ALL arising from measurement downtime, we follow asimilar approach to [83, 86, 87]. The frequencyinstability of the maser can be split into deterministic(e.g. linear drift) and stochastic (e.g. white and pinkfrequency noise) parts. Given the predictable natureof the deterministic part, the resulting offset can bedirectly computed. For example, for the linear driftexhibited by the hydrogen maser, the frequency offsetis the drift rate multiplied by the difference in timebetween the centre of the measurement window andthe average time of the periods in which the clock wasoperational. Following this procedure, we estimate thedeterministic downtime correction based on the driftrate of the hydrogen maser as inferred by the ratio between the OLC and the hydrogen maser during eachmeasurement period.To estimate the uncertainty associated withstochastic fluctuations of the maser, we adopt aMonte-Carlo approach based on simulating month-long hydrogen maser time series using a model ofits frequency noise. For each time series, the offsetbetween the mean frequency during the uptime period T UP and the entire time series T ALL is calculated. Thestandard deviation of these simulated offsets providesan uncertainty estimate for possible frequency errorsarising from the stochastic noise and downtime.The noise model for the hydrogen maser is basedon comparisons against the OLC and is shown in figure3b. It is important to have several extended periodsof measurement to properly capture the behaviour ofthe maser over long periods of measurement downtime.For this campaign, the OLC was operational for 74%of the time with several continuous stretches exceeding strontium optical lattice clock with × − uncertainty and measurement of its absolute frequency Frequency/Hz - 429228004229870
NPL (2017)NICT (2016)PTB (2016)NMIJ (2015)PTB (2015)NICT (2015)SYRTE (2015)NIM (2015)PTB (2014)NMIJ (2013)PTB (2013)NICT (2012)NICT (2011)SYRTE (2011)PTB (2010)UT (2008)JILA (2007)SYRTE (2006)JILA (2006) 10 Fractional Clock Uncertainity
Figure 4.
Left: Summary of all absolute frequencymeasurements of the 5s S to 5s5p P clock transition in Sr since the CIPM first recommended its value in 2006.Measurements were recorded at JILA ( ) [33, 34], Universityof Tokyo ( ) [35], SYRTE ( ) [36–38], PTB ( ) [16, 39, 40, 49],NICT [41–44] ( ), NMIJ [45,46] ( ), NIM [47] ( ), and NPL ( ).Also shown is the updated value for the transition frequency asrecommended by the CIPM in 2017 (blue-shaded region) [14].Right: Contribution from the systematic uncertainty of thestrontium clocks – neglecting gravitational redshift – to the totaluncertainty of each absolute frequency measurement.
24 hours. To reveal the flicker floor of the maser at longtime scales we relaxed our uptime criteria to overlookdowntime periods of up to 2500 seconds, which yieldedseveral quasi-continuous periods extending over severaldays. We found that a noise model comprised of whitephase noise averaging down as 4 × − τ − , combinedwith white frequency noise averaging down as 12 × − τ − / and flicker noise at the 8 × − level,closely approximated the measured maser instabilityfor all observed time scales. Using this model, wesimulated one hundred time series extending overboth measurement periods using Allantools—an opensource software package developed for python [88].Following this procedure, we estimate the uncertaintycontributed by the combination of stochastic masernoise and measurement downtime to be 1 . × − and 2 . × − for the two measurement periods. f UTC(NPL);T all /f TAI;T all
The time offset between UTC(NPL) and UTC iscomputed by the BIPM at 5-day intervals andpublished in the monthly Circular T bulletin. Asour measurement period aligns with these 5-dayintervals, the accumulated time offset can be usedto compute the mean frequency difference betweenour local time scale and TAI over the measurementperiod. The fractional uncertainty associated with thisoffset is calculated based on the type-A link timinguncertainty as specified in Circular T, which for this campaign month was 1 ns. To account for correlationsbetween measurements, we do not directly divide theuncertainty by the total measurement duration T , butinstead extrapolate the error as [89]: u (cid:20) f (UTC(NPL)) f (TAI) − (cid:21) = √ × × (cid:18) T (cid:19) . . (5) f TAI;T all /f SI To complete the evaluation of the secondary frequencystandard against the SI second, the average deviation d over the measurement period of the scale intervalof TAI from the SI second was calculated. Thecomputation of the d values for the two measurementperiods was carried out by the BIPM using the samealgorithm and data that produces the d values reportedin Circular T for each one-month interval of TAI. Forthe first and second measurement intervals the d valueswere 2 . × − and − . × − , respectively.We also account for general-relativistic effectsby transforming from the proper time of the clockto TAI. The relativistic rate shift is computed withrespect to the conventionally adopted equipotential W = 62 636 856 . s − of the Earths gravitypotential. As previously reported in [90], thiscorrection was determined to be − . × − following the procedure outlined in [91] as part of theEMRP project international timescales with opticalclocks (ITOC) [92]. The final absolute frequencymeasurements for both intervals are summarizedin table 2 which also outlines the correction andassociated uncertainty contributed by each frequencyratio in equation 4. Combining the results of the twomeasurements, we estimate the absolute frequency ofthe OLC to be 429 228 004 229 873 . . × − [14], as wellas with measurements at other institutes, as shownin figure 4. Figure 4 also shows the systematicuncertainties of the OLCs at the time that theirabsolute frequencies were reported. Note that theJILA [3] and Tokyo [48] groups have since improvedtheir clocks and both now have predicted uncertaintiesat the low 10 − range.
5. Conclusion
We have realised a strontium optical lattice clockwith an estimated systematic uncertainty of 1 × − ,and determined its absolute frequency by comparisonagainst TAI to be 429 228 004 229 873 . strontium optical lattice clock with × − uncertainty and measurement of its absolute frequency Table 2.
Summary of each of the ratios r specified in equation 4 which are combined to compute the absolute frequency ofthe transition. For each ratio, the fractional deviation from its nominal value r and corresponding uncertainty (68% confidenceinterval) is reported. Fractional values are in units of 1 × − . The value R = 429 228 004 229 873 . S to 5s5p P clock transition in Sr.
Ratio Contribution r Period 1 Period 1 Period 2 Period 2 r/r − r/r − r/r − r/r − f Sr /f UTC(NPL);T UP Ratio at comb R -3917 100 -4437 100Sr statistical – 2 – 3.5Sr systematics 5146 10 5185 10Gravity -1214 4 -1214 4 f UTC(NPL);T UP /f UTC(NPL);T
ALL
Deterministic 1 216 50 119 30Stochastic – 140 – 250 f UTC(NPL);T
ALL /f TAI;T
ALL
Local time scale to TAI 1 154 1200 463 3200 f TAI;T
ALL /f SI TAI to SI second 1 -200 370 90 710 f Sr /f SI Total R
185 1300 206 33001 × − level. As part of an optical fibre networklinking us to other optical clocks around Europe, wewill take part in long-distance comparisons to verifyclock accuracy and test relativistic physics [15, 17]. Toprepare for such measurements, several improvementsto the clock performance are underway: an improved48 .
6. Acknowledgements
The authors thank G´erard Petit for calculating d values for our custom measurement intervals,Rachel Godun and Peter Whibberley for helpfuldiscussions, and Karen Alston and Radka Veltcheva fortemperature sensor calibration. We also note that ourabsolute frequency measurement derives its accuracyfrom primary and secondary standards operated atother national measurement institutes around theworld.This work was financially supported by: theUK Department for Business, Energy and IndustrialStrategy as part of the National MeasurementSystem Programme; the European Metrology ResearchProgramme (EMRP) project SIB55-ITOC; and theEuropean Metrology Programme for Innovation andResearch (EMPIR) project 15SIB03-OC18. This workhas received funding from the EMPIR programmeco-financed by the Participating State and fromthe European Union’s Horizon 2020 research andinnovation programme. The EMRP is jointlyfunded by the EMRP participating countries withinEURAMET and the European Union. W.B. would liketo acknowledge the EU Innovative Training Network (ITN) Future Atomic Clock Technology (FACT). References [1] Huntemann N, Sanner C, Lipphardt B, Tamm C and PeikE 2016
Phys. Rev. Lett. http://link.aps.org/doi/10.1103/PhysRevLett.116.063001 [2] Brewer S M, Chen J S, Hankin A M, Clements E R,Chou C W, Wineland D J, Hume D B and LeibrandtD R 2019
Phys. Rev. Lett. https://link.aps.org/doi/10.1103/PhysRevLett.123.033201 [3] Bothwell T, Kedar D, Oelker E, Robinson J M, Bromley S L,Tew W L, Ye J and Kennedy C J 2019
Metrologia https://doi.org/10.1088%2F1681-7575%2Fab4089 [4] Nemitz N, Ohkubo T, Takamoto M, Ushijima I, DasM, Ohmae N and Katori H 2016 Nature Photonics http://dx.doi.org/10.1038/nphoton.2016.20 [5] McGrew W F, Zhang X, Fasano R J, Sch¨affer S A, Beloy K,Nicolodi D, Brown R C, Hinkley N, Milani G, SchioppoM, Yoon T H and Ludlow A D 2018 Nature https://doi.org/10.1038/s41586-018-0738-2 [6] Heavner T P, Donley E A, Levi F, Costanzo G, ParkerT E, Shirley J H, Ashby N, Barlow S and Jefferts S R2014
Metrologia http://stacks.iop.org/0026-1394/51/i=3/a=174 [7] Li R, Gibble K and Szymaniec K 2011 Metrologia http://stacks.iop.org/0026-1394/48/i=5/a=007 [8] Gu´ena J, Abgrall M, Rovera D, Laurent P, Chupin B, LoursM, Santarelli G, Rosenbusch P, Tobar M, Li R, GibbleK, Clairon A and Bize S 2012 IEEE Transactions onUltrasonics, Ferroelectrics, and Frequency Control Optica [10] Hachisu H, Nakagawa F, Hanado Y and Ido T 2018 Scientific Reports https://doi.org/10.1038/s41598-018-22423-5 [11] Yao J, Sherman J A, Fortier T, Leopardi H, Parker T,McGrew W, Zhang X, Nicolodi D, Fasano R, Sch¨afferS, Beloy K, Savory J, Romisch S, Oates C, DiddamsS, Ludlow A and Levine J 2019 Phys. Rev. Applied strontium optical lattice clock with × − uncertainty and measurement of its absolute frequency https://link.aps.org/doi/10.1103/PhysRevApplied.12.044069 [12] Milner W R, Robinson J M, Kennedy C J, BothwellT, Kedar D, Matei D G, Legero T, Sterr U, RiehleF, Leopardi H, Fortier T M, Sherman J A, LevineJ, Yao J, Ye J and Oelker E 2019 Phys. Rev. Lett. https://link.aps.org/doi/10.1103/PhysRevLett.123.173201 [13] Gill P 2011
Philosophical Transactions of the Royal Societyof London A: Mathematical, Physical and EngineeringSciences
Metrologia http://stacks.iop.org/0026-1394/55/i=2/a=188 [15] Lisdat C, Grosche G, Quintin N, Shi C, Raupach S, GrebingC, Nicolodi D, Stefani F, Al-Masoudi A, D¨orscher S et al. Nature Communications Nature Physics https://doi.org/10.1038/s41567-017-0042-3 [17] Delva P, Lodewyck J, Bilicki S, Bookjans E, Vallet G,Le Targat R, Pottie P E, Guerlin C, Meynadier F,Le Poncin-Lafitte C, Lopez O, Amy-Klein A, Lee W K,Quintin N, Lisdat C, Al-Masoudi A, D¨orscher S, GrebingC, Grosche G, Kuhl A, Raupach S, Sterr U, Hill I R,Hobson R, Bowden W, Kronj¨ager J, Marra G, RollandA, Baynes F N, Margolis H S and Gill P 2017 Phys. Rev.Lett. https://link.aps.org/doi/10.1103/PhysRevLett.118.221102 [18] Chou C, Hume D, Rosenband T and Wineland D 2010
Science [19] Takamoto M, Ushijima I, Ohmae N, Yahagi T, Kokado K,Shinkai H and Katori H 2020
Nature Photonics
URL https://doi.org/10.1038/s41566-020-0619-8 [20] Derevianko A and Pospelov M 2014
Nature Physics http://dx.doi.org/10.1038/nphys3137 [21] Wcis(cid:32)lo P, Ablewski P, Beloy K, Bilicki S, Bober M, BrownR, Fasano R, Ciury(cid:32)lo R, Hachisu H, Ido T et al. Science Advances eaau4869 URL http://dx.doi.org/10.1126/sciadv.aau4869 [22] Huntemann N, Lipphardt B, Tamm C, Gerginov V, WeyersS and Peik E 2014 Phys. Rev. Lett. http://link.aps.org/doi/10.1103/PhysRevLett.113.210802 [23] Godun R M, Nisbet-Jones P B R, Jones J M, King S A,Johnson L A M, Margolis H S, Szymaniec K, Lea S N,Bongs K and Gill P 2014
Phys. Rev. Lett. http://link.aps.org/doi/10.1103/PhysRevLett.113.210801 [24] Ye J, Kimble H and Katori H 2008
Science [25] Yasuda M, Inaba H, Kohno T, Tanabe T, Nakajima Y,Hosaka K, Akamatsu D, Onae A, Suzuyama T, AmemiyaM and Hong F L 2012
Applied Physics Express https://doi.org/10.1143%2Fapex.5.102401 [26] Hinkley N, Sherman J A, Phillips N B, Schioppo M, LemkeN D, Beloy K, Pizzocaro M, Oates C W and Ludlow A D2013 Science https://science.sciencemag.org/content/341/6151/1215 [27] Kim H, Heo M S, Lee W K, Park C Y, Hong H G,Hwang S W and Yu D H 2017
Japanese Journal ofApplied Physics https://doi.org/10.7567%2Fjjap.56.050302 [28] Pizzocaro M, Bregolin F, Barbieri P, Rauf B, Levi F and Calonico D 2019 Metrologia
URL https://iopscience.iop.org/article/10.1088/1681-7575/ab50e8/meta [29] Yamanaka K, Ohmae N, Ushijima I, Takamoto M andKatori H 2015
Phys. Rev. Lett. http://link.aps.org/doi/10.1103/PhysRevLett.114.230801 [30] Tyumenev R, Favier M, Bilicki S, Bookjans E, Le TargatR, Lodewyck J, Nicolodi D, Le Coq Y, Abgrall M,Gu´ena J et al.
New Journal of Physics https://doi.org/10.1088%2F1367-2630%2F18%2F11%2F113002 [31] Kaneda Y, Yarborough J M, Merzlyak Y, YamaguchiA, Hayashida K, Ohmae N and Katori H 2016 Opt.Lett. http://ol.osa.org/abstract.cfm?URI=ol-41-4-705 [32] Kulosa A P, Fim D, Zipfel K H, R¨uhmann S, Sauer S, JhaN, Gibble K, Ertmer W, Rasel E M, Safronova M S,Safronova U I and Porsev S G 2015 Phys. Rev. Lett. http://link.aps.org/doi/10.1103/PhysRevLett.115.240801 [33] Boyd M M, Ludlow A D, Blatt S, Foreman S M, Ido T,Zelevinsky T and Ye J 2007
Phys. Rev. Lett. http://link.aps.org/doi/10.1103/PhysRevLett.98.083002 [34] Campbell G, Ludlow A, Blatt S, Thomsen J, Martin M, DeMiranda M, Zelevinsky T, Boyd M, Ye J, Diddams S,Heavner T, Parker T and Jefferts S 2008 Metrologia http://iopscience.iop.org/0026-1394/45/5/008/ [35] Hong F L, Musha M, Takamoto M, Inaba H, YanagimachiS, Takamizawa A, Watabe K, Ikegami T, Imae M, FujiiY, Amemiya M, Nakagawa K, Ueda K and Katori H2009 Opt. Lett. http://ol.osa.org/abstract.cfm?URI=ol-34-5-692 [36] Baillard X, Fouch´e M, Le Targat R, WestergaardP, Lecallier A, Le Coq Y, Rovera G, Bize Sand Lemonde P 2007 Opt. Lett. [37] Le Targat R, Lorini L, Le Coq Y, Zawada M, Gu´enaJ, Abgrall M, Gurov M, Rosenbusch P, Rovera D G,Nag´orny B, Gartman R, Westergaard P G, TobarM E, Lours M, Santarelli G, Clairon A, Bize S,Laurent P, Lemonde P and Lodewyck J 2013 NatureCommunications URL http://dx.doi.org/10.1038/ncomms3109 [38] Lodewyck J, Bilicki S, Bookjans E, Robyr J L, Shi C, ValletG, LeTargat R, Nicolodi D, Coq Y L, Gu´ena J, AbgrallM, Rosenbusch P and Bize S 2016
Metrologia http://stacks.iop.org/0026-1394/53/i=4/a=1123 [39] Falke S, Schnatz H, Winfred J S R V, Middelmann T,Vogt S, Weyers S, Lipphardt B, Grosche G, Riehle F,Sterr U and Lisdat C 2011 Metrologia http://stacks.iop.org/0026-1394/48/i=5/a=022 [40] Falke S, Lemke N, Grebing C, Lipphardt B, Weyers S,Gerginov V, Huntemann N, Hagemann C, Al-MasoudiA, H¨afner S, Vogt S, Sterr U and Lisdat C 2014 NewJournal of Physics http://stacks.iop.org/1367-2630/16/i=7/a=073023 [41] Matsubara K, Li Y, Nagano S, Ito H, Kajita M, Kojima R,Hayasaka K, Hanado Y and Hosokawa M 2009 Absolutefrequency measurement of the Ca + clock transitionusing a LD-based clock laser and UTC(NICT) The 2009Joint Meeting of the European Frequency and TimeForum and the IEEE International Frequency ControlSymposium pp 751 –755[42] Yamaguchi A, Fujieda M, Kumagai M, Hachisu H, NaganoS, Li Y, Ido T, Takano T, Takamoto M and KatoriH 2011
Applied Physics Express http://stacks.iop.org/1882-0786/4/i=8/a=082203 strontium optical lattice clock with × − uncertainty and measurement of its absolute frequency [43] Hachisu H, Petit G and Ido T 2016 Applied Physics B
34 URL https://doi.org/10.1007/s00340-016-6603-9 [44] Hachisu H, Petit G, Nakagawa F, Hanado Y and IdoT 2017
Opt. Express [45] Akamatsu D, Inaba H, Hosaka K, Yasuda M, Onae A,Suzuyama T, Amemiya M and Hong F L 2014 AppliedPhysics Express http://stacks.iop.org/1882-0786/7/i=1/a=012401 [46] Tanabe T, Akamatsu D, Kobayashi T, Takamizawa A,Yanagimachi S, Ikegami T, Suzuyama T, Inaba H, OkuboS, Yasuda M, Hong F L, Onae A and Hosaka K 2015 Journal of the Physical Society of Japan https://doi.org/10.7566/JPSJ.84.115002 [47] Lin Y G, Wang Q, Li Y, Meng F, Lin B K, Zang E J,Sun Z, Fang F, Li T C and Fang Z J 2015 ChinesePhysics Letters https://doi.org/10.1088%2F0256-307x%2F32%2F9%2F090601 [48] Ushijima I, Takamoto M, Das M, Ohkubo T and Katori H2015 Nature Photonics http://dx.doi.org/10.1038/nphoton.2015.5 [49] Koller S B, Grotti J, Vogt S, Al-Masoudi A, D¨orscherr S,H¨afner S, Sterr U and Lisdat C 2017 Phys. Rev. Lett. https://link.aps.org/doi/10.1103/PhysRevLett.118.073601 [50] Hill I R, Hobson R, Bowden W, Bridge E M, DonnellanS, Curtis E A and Gill P 2016
Journal of Physics:Conference Series http://stacks.iop.org/1742-6596/723/i=1/a=012019 [51] Donnellan S, Hill I R, Bowden W and Hobson R 2019
Review of Scientific Instruments https://aip.scitation.org/doi/10.1063/1.5051124 [52] Hill I R, Ovchinnikov Y B, Bridge E M, Curtis E A andGill P 2014 Journal of Physics B: Atomic, Molecularand Optical Physics http://stacks.iop.org/0953-4075/47/i=7/a=075006 [53] Hobson R, Bowden W, Vianello A, Hill I R and Gill P 2020 Physical Review A https://link.aps.org/doi/10.1103/PhysRevA.101.013420 [54] Katori H, Ido T, Isoya Y and Kuwata-Gonokami M 1999
Phys. Rev. Lett. http://link.aps.org/doi/10.1103/PhysRevLett.82.1116 [55] Loftus T, Ido T, Boyd M, Ludlow A and Ye J 2004 Phys.Rev. A http://link.aps.org/doi/10.1103/PhysRevA.70.063413 [56] Mukaiyama T, Katori H, Ido T, Li Y and Kuwata-Gonokami M 2003 Phys. Rev. Lett. http://link.aps.org/doi/10.1103/PhysRevLett.90.113002 [57] Ludlow A, Huang X, Notcutt M, Zanon-Willette T,Foreman S, Boyd M, Blatt S and Ye J 2007 Opt.Lett. [58] Hagemann C, Grebing C, Kessler T, Falke S, Lemke N,Lisdat C, Schnatz H, Riehle F and Sterr U 2013 IEEETransactions on Instrumentation and Measurement AppliedPhysics B http://dx.doi.org/10.1007/s003400100735 [60] Safronova M S, Porsev S G, Safronova U I, Kozlov M G andClark C W 2013 Phys. Rev. A http://link.aps.org/doi/10.1103/PhysRevA.87.012509 [61] Porsev S and Derevianko A 2006 Phys. Rev. A http://link.aps.org/doi/10.1103/PhysRevA.74.020502 [62] Middelmann T, Falke S, Lisdat C and Sterr U 2012 Phys.Rev. Lett. http://link.aps.org/doi/10.1103/PhysRevLett.109.263004 [63] Nicholson T L, Campbell S L, Hutson R B, Marti G E,Bloom B J, McNally R L, Zhang W, Barrett M D,Safronova M S, Strouse G F, Tew W L and Ye J 2015
Nature Communications URL http://dx.doi.org/10.1038/ncomms7896 [64] Preston-Thomas H 1990
Metrologia http://stacks.iop.org/0026-1394/27/i=1/a=002 [65] BIPM, IFCC, IUPAC and ISO 2008 JCGM 100:2008 [66] Martin B J 2014
Building a Better Atomic Clock
Ph.D.thesis University of Colorado[67] Middelmann T, Lisdat C, Falke S, Winfred J, Riehle F andSterr U 2011
IEEE Transactions on Instrumentation andMeasurement The European Physical Journal D - Atomic, Molecular,Optical and Plasma Physics http://dx.doi.org/10.1140/epjd/e2007-00330-3 [69] Boyd M, Zelevinsky T, Ludlow A, Blatt S, Zanon-WilletteT, Foreman S and Ye J 2007 Phys. Rev. A Journal of the Physical Society of Japan http://jpsj.ipap.jp/link?JPSJ/75/104302/ [71] Brown R C, Phillips N B, Beloy K, McGrew W F, SchioppoM, Fasano R J, Milani G, Zhang X, Hinkley N, LeopardiH, Yoon T H, Nicolodi D, Fortier T M and LudlowA D 2017 Phys. Rev. Lett. https://link.aps.org/doi/10.1103/PhysRevLett.119.253001 [72] Westergaard P, Lodewyck J, Lorini L, Lecallier A, BurtE, Zawada M, Millo J and Lemonde P 2011
Phys. Rev.Lett. http://dx.doi.org/10.1103/PhysRevLett.106.210801 [73] Blatt S, Thomsen J W, Campbell G K, Ludlow A D,Swallows M D, Martin M J, Boyd M M and Ye J 2009
Phys. Rev. A http://link.aps.org/doi/10.1103/PhysRevA.80.052703 [74] Gibble K 2013 Phys. Rev. Lett. http://link.aps.org/doi/10.1103/PhysRevLett.110.180802 [75] Mitroy J and Zhang J 2010
Molecular Physics http://dx.doi.org/10.1080/00268976.2010.501766 [76] Bowden W, Hill I R, Baird P E G and Gill P 2017
Reviewof Scientific Instruments https://doi.org/10.1063/1.4973774 [77] D¨orscher S, Schwarz R, Al-Masoudi A, Falke S, Sterr Uand Lisdat C 2018 Phys. Rev. A https://link.aps.org/doi/10.1103/PhysRevA.97.063419 [78] Hill I R 2012 Development of an apparatus for a strontiumoptical lattice optical frequency standard
Ph.D. thesisImperial College London[79] Ma L S, Jungner P, Ye J and Hall J 1994
Opt.Lett. [80] Park C, Yu D, Lee W, Park S, Kim E, Lee S, Cho J, Yoon T,Mun J, Park S, Kwon T and Lee S 2013 Metrologia https://doi.org/10.1088%2F0026-1394%2F50%2F2%2F119 [81] Huang Y, Guan H, Liu P, Bian W, Ma L, Liang K, Li T andGao K 2016 Phys. Rev. Lett. https://link.aps.org/doi/10.1103/PhysRevLett.116.013001 [82] Dub´e P, Bernard J E and Gertsvolf M 2017
Metrologia https://doi.org/10.1088%2F1681-7575%2Faa5e60 [83] Baynham C F A, Godun R M, Jones J M, King S A,Nisbet-Jones P B R, Baynes F, Rolland A, Baird P E G,Bongs K, Gill P and Margolis H S 2018 Journal ofModern Optics https://doi.org/10. strontium optical lattice clock with × − uncertainty and measurement of its absolute frequency [84] McGrew W F, Zhang X, Leopardi H, Fasano R J, NicolodiD, Beloy K, Yao J, Sherman J A, Sch¨affer S A,Savory J, Brown R C, R¨omisch S, Oates C W, ParkerT E, Fortier T M and Ludlow A D 2019 Optica [85] Petit G and Panfilo G 2018 Optimal traceability to the SIsecond through TAI Proceedings of the 2018 EuropeanFrequency and Time Forum (EFTF) pp 185–187[86] Hachisu H and Ido T 2015
Japanese Journal of Ap-plied Physics http://stacks.iop.org/1347-4065/54/i=11/a=112401 [87] Yu D H, Weiss M and Parker T E 2007 Metrologia https://doi.org/10.1088%2F0026-1394%2F44%2F1%2F014 [88] Wallin A E E, Price D C, Carson C G and Meynadier F2018 allantools: Allan deviation calculation URL https://ui.adsabs.harvard.edu/abs/2018ascl.soft04021W [89] Panfilo G and Parker T E 2010 Metrologia http://stacks.iop.org/0026-1394/47/i=5/a=005 [90] Riedel F, Al-Masoudi A, Benkler E, Drscher S, Gerginov V,Grebing C, Hfner S, Huntemann N, Lipphardt B, LisdatC, Peik E, Piester D, Sanner C, Tamm C, Weyers S,Denker H, Timmen L, Voigt C, Calonico D, Cerretto G,Costanzo G A, Levi F, Sesia I, Achkar J, Guna J, AbgrallM, Rovera G D, Chupin B, Shi C, Bilicki S, Bookjans E,Lodewyck J, Targat R L, Delva P, Bize S, Baynes F N,Baynham C, Bowden W, Gill P, Godun R M, Hill I R,Hobson R, Jones J M, King S A, Nisbet-Jones P, RollandA, Shemar S L, Whibberley P B and Margolis H S 2020 Metrologia
URL http://iopscience.iop.org/10.1088/1681-7575/ab6745 [91] Denker H, Timmen L, Voigt C, Weyers S, Peik E, MargolisH S, Delva P, Wolf P and Petit G 2018
Journal ofGeodesy https://doi.org/10.1007/s00190-017-1075-1 [92] Margolis H S, Godun R M, Gill P, Johnson L A M, ShemarS L, Whibberley P B, Calonico D, Levi F, Lorini L,Pizzocaro M, Delva P, Bize S, Achkar J, Denker H,Timmen L, Voigt C, Falke S, Piester D, Lisdat C, SterrU, Vogt S, Weyers S, Gersl J, Lindvall T and Merimaa M2013 International timescales with optical clocks (ITOC) Proceedings of the 2013 Joint Meeting of the EuropeanFrequency and Time Forum and the IEEE InternationalFrequency Control Symposium pp 908–911[93] H¨afner S, Falke S, Grebing C, Vogt S, Legero T, Merimaa M,Lisdat C and Sterr U 2015
Opt. Lett. http://ol.osa.org/abstract.cfm?URI=ol-40-9-2112 [94] Ushijima I, Takamoto M and Katori H 2018 Phys. Rev.Lett.263202 URL