Least-squares analysis of clock frequency comparison data to deduce optimized frequency and frequency ratio values
aa r X i v : . [ phy s i c s . d a t a - a n ] A p r Least-squares analysis of clock frequencycomparison data to deduce optimized frequency andfrequency ratio values
H. S. Margolis and P. Gill
National Physical Laboratory, Teddington, Middlesex TW11 0LW, UKE-mail: [email protected]
Abstract.
A method is presented for analysing over-determined sets of clockfrequency comparison data involving standards based on a number of differentreference transitions. This least-squares adjustment procedure, which is basedon the method used by CODATA to derive a self-consistent set of values for thefundamental physical constants, can be used to derive optimized values for thefrequency ratios of all possible pairs of reference transitions. It is demonstratedto reproduce the frequency values recommended by the International Committeefor Weights and Measures, when using the same input data used to derive thosevalues. The effects of including more recently published data in the evaluationis discussed and the importance of accounting for correlations between the inputdata is emphasised.
1. Introduction
The most advanced optical frequency standards have now reached levels of stabilityand accuracy [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11] that significantly surpass the performanceof the best caesium primary standards [12, 13, 14, 15, 16], raising the possibilityof a future redefinition of the SI second [17]. As a first step towards preparing for aredefinition the International Committee for Weights and Measures (CIPM), followinga recommendation of the Consultative Committee for Time and Frequency (CCTF),introduced the concept of secondary representations of the second that may be used torealize the second in parallel to the caesium primary standard. Seven optical frequencystandards (and one microwave frequency standard) may currently be used as secondaryrepresentations of the second, with recommended frequencies and uncertainties beingassigned by the Frequency Standards Working Group (WGFS) of the CCTF andConsultative Committee for Length (CCL). These recommended frequency values areperiodically updated and published on the website of the International Bureau ofWeights and Measures [18].With a single exception, the data considered so far by the WGFS comes fromabsolute frequency measurements made relative to caesium fountain primary frequencystandards. However future information about the reproducibility of the opticalstandards will come mainly from direct optical frequency ratio measurements [19,20, 21, 8]. For example, within the EMRP-funded International Timescales withOptical Clocks (ITOC) project [22], a coordinated comparison programme will leadto a set of frequency ratio measurements between high accuracy optical frequency east-squares analysis of clock frequency comparison data ν a /ν b could be measureddirectly or it could be determined indirectly by combining two or more other frequencyratio measurements (e.g. ν a /ν b = ( ν a /ν c )( ν c /ν b ) or ν a /ν b = ( ν a /ν d )( ν d /ν c )( ν c /ν b )).If the indirect determinations of this frequency ratio have comparable uncertaintiesto the direct determination then all of them must be considered in deriving a “best”value for the frequency ratio. These multiple routes to deriving each frequency ratiovalue will complicate the task of the WGFS because it will no longer be possible totreat each optical frequency standard in isolation when considering the available data.Here we describe a possible approach to analyzing over-determined sets offrequency comparison data to deduce optimized values for the frequency ratios betweeneach contributing standard. The paper is organised as follows. In section 2 wedescribe the analysis procedure, which follows a least-squares approach. The testswe have carried out to ensure that the algorithms have been correctly implementedin our software are described in section 3. In section 4 we consider the body offrequency comparison data presently available in the published literature, and discusshow the CIPM recommended frequency values might change if recent measurementswere included in the evaulation. However this analysis neglects correlations betweenthe individual measurements, and in section 5 we discuss how this affects the resultsobtained. Finally, section 6 contains some conclusions and perspectives that may berelevant for future discussions within the WGFS.
2. Analysis procedure
To derive a self-consistent set of optimised frequency ratio values from a set of clockfrequency comparison experiments we use a least-squares adjustment procedure. Thisis based on the approach used by the Committee on Data for Science and Technology(CODATA) to provide a self-consistent set of internationally recommended values ofthe fundamental physical constants [23]. Our method is illustrated in figure 1.Suppose that the frequency standards involved in the comparison experimentsare based on N S different reference transitions, with frequencies ν k ( k = 1 , , . . . , N S ).For example ν could be the 5s S –5s5p P transition in Sr at 698 nm, ν the6s S / –4f F / transition in Yb + at 467 nm, ν the 9.2 GHz 6s S / ( F = 3)–6s S / ( F = 4) ground-state hyperfine transition in Cs, and so on. The set offrequency comparison experiments yields a set of N measured values q i of variousquantities, which may be optical frequency ratios, microwave frequency ratios, oroptical-microwave frequency ratios. These measured values q i , together with theirstandard uncertainties u i , variances u i = u ii and covariances u ij (where u ji = u ij ),form the input data to the least-squares adjustment.As the first step, we choose a set of M = N S − z j (where M < N ) given by z j = ν j ν j +1 , (1)where j = 1 , , . . . , N S −
1. These adjusted frequency ratios are the variables in theleast-squares adjustment and are equivalent to the adjusted constants in the CODATA east-squares analysis of clock frequency comparison data Input: Set of N measured frequency ratios q i , variances u i and covariances u ij Choose set of M = N S - 1 adjusted frequency ratios z j Express measured frequency ratios in terms of adjusted frequency ratios, yielding a set of N equationsInput: Initial estimates of adjusted frequency ratios Linearize equations using a Taylor expansion around initial estimates of adjusted frequency ratios s j Perform least-squares adjustmentOutput: Best values of adjusted frequency ratios, variances and covariancesAre adjusted frequency ratios sufficiently close to initial estimates? Use output from least-squares adjustment as new starting valuesCalculate other frequency ratios and uncertainties from adjusted frequency ratios and their covariance matrixPerform self-consistency checks Output: Optimized frequency ratios(including absolute frequencies)Output: Birge ratio and normalized residualsyes no
Figure 1.
Analysis procedure used to derive a self-consistent set of optimizedfrequency ratio values from a set of frequency comparison experiments involvingstandards based on N S different reference transitions. analysis of the fundamental physical constants. Our choice of z j meets the necessarycondition that no adjusted frequency ratio may be expressed as a function of theothers.The quantities q i that are measured in the frequency comparison experiments arenext expressed as a function f i of one or more of these adjusted frequency ratios z j by the set of N equations q i . = f i ( z , z , . . . , z M ) (2)where i = 1 , , . . . , N . Here the symbol . = is used to indicate that in general the leftand right hand sides of the equation are not exactly equal, because the set of equationsis overdetermined. For example, the first observed quantity q might be ν /ν , whichcan be expressed as z z z , while the second observed quantity q might be eitheranother measurement of ν /ν or a measurement of a different ratio such as ν /ν .The equations (2) are, in most cases, nonlinear. Prior to the least-squaresadjustment, they are therefore linearized using a Taylor expansion around startingvalues s j (initial estimates of the adjusted frequency ratios): q i . = f i ( s , s , . . . , s M )+ M X j =1 ∂f i ( s , s , . . . , s M ) ∂s j ( z j − s j ) + · · · . (3)This enables linear matrix methods to be applied. Defining new variables y i = q i − f i ( s , s , . . . , s M ) (4) east-squares analysis of clock frequency comparison data x j = z j − s j , (5)the linearized equations (3) can be rewritten in the form y i . = M X j =1 a ij x j (6)where a ij = ∂f i ( s , s , . . . , s M ) ∂s j . (7)In matrix notation these equations become Y . = AX , (8)where Y is a column matrix with N elements, A is a rectangular matrix with N rows and M columns, and X is a column matrix with M elements. In the sameway, we consider the measured frequency ratios q i , the functions f i and the adjustedfrequency ratios z j and their initial estimates s j to be elements of matrices Q , F , Z and S respectively.To obtain the best estimate of X , and hence of the adjusted frequency ratios Z ,a least-squares adjustment is performed, minimizing the product S = ( Y − AX ) T V − ( Y − AX ) (9)with respect to X . Here V = cov Y is the N × N covariance matrix of the input data,with elements u ij . The solution ˆ X is used to calculate the best estimate ˆ Z of theadjusted frequency ratios according toˆ Z = S + ˆ X . (10)Although the solution of the linear approximation (3) does not provide an exactsolution of the nonlinear equations (2), the values of the adjusted frequency ratios ˆ Z obtained from the least-squares adjustment will be an improvement over the startingvalues S . To obtain more precise values, these improved values of the adjustedfrequency ratios are used as starting values for a new linear approximation and asecond least-squares adjustment is performed. This procedure is repeated until thenew values of the adjusted frequency ratios obtained from the least-squares adjustmentdiffer from the starting values for that iteration by a sufficiently small fraction of theiruncertainties. The number of iterations required to satisfy this condition will dependon how close the starting values s j are to the values of ˆ z j calculated in the finaliteration. Once convergence has been achieved, the best estimate ˆ Q of the measuredquantities Q can be calculated from the final solution ˆ X :ˆ Q = F + A ˆ X . (11)In general, the values of the adjusted frequency ratios obtained from the least-squares adjustment will be correlated. The covariance matrix cov( ˆ Z ) = cov( ˆ X ),whose elements u (ˆ z i , ˆ z j ) are the covariances of the adjusted frequency ratios, can beused to evaluate the uncertainty in other frequency ratios calculated from two ormore of the adjusted frequency ratios. In general, there will be a total of K possiblefrequency ratios ˆ r i which may be expressed in terms of the adjusted frequency ratiosˆ z j : ˆ r i (ˆ z , ˆ z , . . . , ˆ z M ) (12) east-squares analysis of clock frequency comparison data i = 1 , , . . . , K , including expressions of the form ˆ r i = ˆ z j . According to thestandard formula for the propagation of uncertainty, the covariances of these optimizedfrequency ratios are given by u (ˆ r k , ˆ r l ) = M X i,j =1 ∂ ˆ r k ∂ ˆ z i ∂ ˆ r l ∂ ˆ z j u (ˆ z i , ˆ z j ) . (13)If l = k , then equation (13) gives the variances u (ˆ r k ) = u (ˆ r k , ˆ r k ) .Self-consistency checks on the body of data provide verification of the uncertaintyevaluations for each individual frequency standard and enable any issues withindividual frequency standards to be identified. To obtain a measure of the consistencyof the input data, the Birge ratio R B = (cid:18) χ N − M (cid:19) / , (14)is computed. Here χ = ( Q − ˆ Q ) T V − ( Q − ˆ Q ) , (15)is the minimum value of S as given by equation (9), evaluated in the final iterationof the least-squares adjustment. A Birge ratio significantly larger than one suggeststhat the input data are inconsistent. Similarly, a normalized residual ρ i = q i − ˆ q i u i (16)significantly larger than one for a particular measured frequency ratio q i suggests thatthe measurement is inconsistent with the other data.These algorithms have been implemented in Matlab, with the least-squaressolution to equation (8) being found using the routine lscov() . Due to theextremely high accuracy with which frequency comparisons can be performed,numerical calculations must be performed with a precision of more than 18 significantfigures. This is achieved using routines designed for high precision floating pointarithmetic [24].
3. Tests of the software algorithms
Several simulated test data sets were generated for testing the analysis software. Thesewere generated in such a way that the software, if functioning correctly, would generatecertain known values for the absolute frequencies of each reference transition. Sincethe software does not calculate these frequencies or their uncertainties directly, butinstead calculates them from combinations of the adjusted frequency ratios and thecovariance matrix determined from the least-squares adjustment, this approach isconsidered to provide a good check that the adjusted frequency ratios are also beingcalculated correctly.One key test of the software algorithms employed in the analysis procedure iswhether they are able to reproduce the CIPM recommended frequency values, whenusing the same input data employed by the CCL-CCTF WGFS. To check this, the setof N S = 15 frequency standards listed in table 1 was considered.The results obtained for the seven optical secondary representations of the secondare shown in figure 2. All these frequency values agree with the CIPM values, butthe uncertainties determined from the least-squares adjustment procedure are smaller east-squares analysis of clock frequency comparison data Table 1.
Frequency standards considered in the tests of the least-squaresadjustment procedure.Atom/ion Reference transition In + S –5s5p P H 1s S / –2s S / Hg 6s S –6s6p P Al + S –3s3p P Hg + S / –5d D / Yb + S / –5d D / Yb + S / –4f F / Yb 6s S –6s6p P Ca 4s S –4s4p P Sr + S / –4d D / Sr 5s S –5s5p P Sr 5s S –5s5p P Ca + S / –3d D / Rb 5s S / ( F = 1)–5s S / ( F = 2) Cs 6s S / ( F = 3)–6s S / ( F = 4) than the uncertainties assigned by the CIPM, sometimes by a factor of two or three.The explanation for this is that the WGFS takes a conservative approach to estimatinguncertainties, typically multiplying the relative standard uncertainty on the weightedmean of a set of frequency values by a factor of two or three to reflect the fact thatthe measurements originate from only a few independent research groups (or in somecases a single research group). Our analysis procedures yield uncertainties equivalentto the relative standard uncertainty.The frequency values we obtain for the other standards listed in table 1 are insimilarly good agreement with the CIPM values, with the sole exception of the Caoptical frequency standard. The reason for this discrepancy is that in this case theWGFS departed from its normal procedure of taking a weighted mean of the availablefrequency measurements. Instead, bearing in mind a significant discrepancy betweentwo independent frequency measurements, an unweighted mean of the two values wasused. Our analysis procedures, on the other hand, are equivalent to taking a weightedmean of the two values.
4. Inclusion of new frequency comparison data
Since the last review of available frequency data by the WGFS, which was performedin September 2012, a number of new frequency comparison results have been reported.These include ten new absolute frequency measurements (one of the 1S–2S transitionin H [25], two each of the E2 [26, 8] and E3 [8, 9] reference transitions in Yb + , oneeach of the reference transitions in Yb [27] and Sr + [6] and three of the referencetransition in Sr [3, 28, 7]). However three optical frequency ratios have also beenmeasured directly. These are the ratios between the reference transitions in Ca + and Sr [20], between the reference transitions in
Yb and Sr [21], and betweenthe E2 and E3 reference transitions in Yb + [8].The effects on the optimized frequency values obtained for the optical secondaryrepresentations of the second of including this additional data in the analysis (but atthis stage ignoring any correlations between the input data) are shown in figure 2. east-squares analysis of clock frequency comparison data -3-2-10123 Sr +171 Yb Yb + E3 Sr Yb + E2 Al + ( ν - ν C I P M ) / H z Hg + Figure 2.
Frequency values obtained for the seven optical secondaryrepresentations of the second, calculated using the same input data used by theCCL-CCTF WGFS (green triangles) and with new data included in the analysis(red circles). The present CIPM recommended frequency values are also shown(blue squares).
Significant changes are observed for some of the optimized frequency values, mostnotably the Yb + and Sr + optical frequency standards, but also for Sr.This illustrates that the conservative approach adopted by the WGFS in assigninguncertainties to the recommended frequency values may indeed be prudent.The Sr data is, however, coupled to the data for Ca + by the optical frequencyratio measurement between these two reference transitions [20], and the presentlyavailable body of data for Ca + is not internally self-consistent. To assess howmuch effect this coupling has on the optimized frequency value for the Sr referencetransition, the Ca + / Sr frequency ratio measurement was removed from the inputdata and the least-squares adjustment repeated. The difference from the currentCIPM recommended frequency value for Sr was found to reduce only slightly from0.28 Hz to 0.25 Hz.
5. Importance of correlations
The above analysis neglects any correlations between the measured frequency ratiovalues used as input to the least-squares adjustment. In reality, there will becorrelations among the input data that should be accounted for. For example, atNPL, the absolute frequencies of the E2 and E3 reference transitions in Yb + andthe direct optical frequency ratio between them were recently all determined during asingle measurement campaign, with substantial periods of overlap in the data-takingperiods [8]. This means that there are significant correlations between these threevalues. As frequency comparison experiments involving multiple optical frequencystandards and even multiple laboratories become more frequent, such correlationswill become more frequent and more significant. However the covariances and east-squares analysis of clock frequency comparison data Operational standards durin g 10-day measurement campaignDay1 2 3 4 5 6 7 8Cs Sr +171 Yb + E2 Yb + E3Measurements made during campaign Yb + E2 / Cs Yb + E3 / Cs Sr + / Cs Yb + E2 / Yb + E3 Yb + E2 / Sr + 171 Yb + E3 / Sr + Figure 3.
Hypothetical 10-day measurement campaign involving four differentfrequency standards. The top half of the figure shows which standards areoperating during each day of the campaign, whilst the lower half shows whichfrequency ratios can be determined from the data taken each day. corresponding correlation coefficients between different frequency ratio measurementsare not normally reported in the literature, even for values obtained in the samelaboratory. To evaluate these covariances additional information would typically needto be obtained from each research group involved, just as in the CODATA least-squaresadjustment of the fundamental physical constants [23].To illustrate the importance of properly accounting for correlations, we considerthe hypothetical 10-day measurement campaign illustrated in figure 3. This involvesa caesium primary frequency standard, which we assume operates 100% of the time,and three optical frequency standards, which we assume each run for 60% of the time,with some periods of overlap. Six different frequency ratios can be determined, eachfrom different periods of the campaign. For these six frequency ratios, there are twelvenon-zero correlation coefficients.Correlations arise from both statistical and systematic uncertainties. Forexample, the absolute frequency measurement of the E2 transition in Yb + and theabsolute frequency measurement of the reference transition in Sr + are correlatedbecause part of the caesium primary frequency standard data is common to thetwo. Assuming initially that all other sources of uncertainty are negligible comparedto the statistical uncertainty associated with the caesium standard, then since the Yb + E2 standard runs for a total period T A = 6 days and the Sr + standardruns for a total period T B = 6 days, with a period of overlap T overlap = 3 days, wecalculate a correlation coefficient of q T / ( T A T B ) = 0 .
5. However in practice fora measurement of this duration, our initial assumption is not a good one and othercontributions to the uncertainty must be considered. For this particular example, there east-squares analysis of clock frequency comparison data -1.5-1.0-0.50.00.51.01.5 ( ν - ν C I P M ) / H z Yb + E2 Yb + E3 Sr + Figure 4.
Effect of correlations on the output from the least-squares adjustmentfor the hypothetical measurement campaign illustrated in figure 3. Blue trianglesshow the absolute frequency values obtained when correlations are neglected,whilst red squares show the corresponding values obtained when correlations areincluded. is an additional source of correlation because the systematic uncertainty of the caesiumfountain is also common to the two measurements. In a similar way, correlationsbetween the absolute frequency measurement of the E3 transition in Yb + andthe Yb + E2 / Yb + E3 frequency ratio measurement arise from both the statisticaland the systematic uncertainty of the Yb + E3 standard. If we use the presentstabilities and systematic uncertainties of NPL’s frequency standards [6, 8, 15] toestimate the correlation coefficients for this hypothetical measurement campaign, wefind that the values of the twelve correlation coefficients range from − .
102 to 0 . + E2 / Yb + E3 and Yb + E2 / Sr + frequency ratios, since the uncertainties of both measurements would be dominated bythe systematic uncertainty of the Yb + E2 standard at its current state of development.For arbitrarily-selected values of the measured frequency ratios resulting from thishypothetical 10-day measurement campaign, the effect of correlations on the optimizedfrequency ratios and absolute frequencies can be determined. As illustrated in figure 4,neglecting correlations leads to too much weight being given to these measurements inthe least-squares adjustment, resulting in biased frequency values and underestimateduncertainties. For some planned measurement campaigns, even stronger correlationscould potentially arise, resulting in more significant biases if these correlations wereneglected in the analysis. This demonstrates the importance of gathering informationabout the correlations between the input data, both for intra-laboratory frequencycomparisons and for inter-laboratory frequency comparisons. east-squares analysis of clock frequency comparison data
6. Conclusion
In summary, we have described an analysis procedure that can be applied todetermine a self-consistent set of frequency ratios between high accuracy frequencystandards (both optical and microwave) based on all experimental data availableat any particular time, and including correlations among the data. Currently thematrix of frequency comparison data is rather sparsely populated, but as the numberof frequency ratio measurements made without reference to the caesium primarystandard increases, our methods and software could be used to provide valuableinformation about the relative performance of different candidates for an opticalredefinition of the SI second. They can also be used to determine optimized valuesand uncertainties for the absolute frequencies of each optical standard relative to thecurrent definition of the SI second, since these are simply special cases of frequencyratios involving the caesium primary standard. Such absolute frequency values willbe required to maximize the potential contribution of optical clocks to internationaltimescales prior to any redefinition.Our work also identifies the key issues likely to be encountered in assessingan over-determined set of frequency comparison data. Firstly, it will be necessaryto identify and critically review all possible input data, with particular attentionbeing paid to the standard uncertainty associated with each measurement. Secondly,the correlations between the input data must be considered, since these can have asignificant effect on the frequency ratio values and uncertainties obtained from theleast-squares adjustment. However the information reported in the literature is inmany cases insufficient to calculate the correlation coefficients, meaning that it willbe necessary to seek additional information from the groups that carried out themeasurements in order that each input datum is given the appropriate weight in theleast-squares adjustment. Finally, it will be necessary to investigate the extent towhich each input datum contributes to the determination of the adjusted frequencyratios as well as the effects of omitting inconsistent or inconsequential data as maybe deemed appropriate. None of these issues are unique to the problem discussedhere; indeed they are common to those that have been faced by the CODATA TaskGroup on Fundamental Constants for many years. As such they are likely to be highlyrelevant to future discussions within the CCL-CCTF Frequency Standards WorkingGroup.
Acknowledgments
This work was performed within the ITOC project as part of the EuropeanMetrology Research Programme (EMRP). The EMRP is jointly funded by theEMRP participating countries within EURAMET and the European Union. Theauthors gratefully acknowledge helpful discussions with partners in the ITOC projectconsortium and with members of the CCL-CCTF WGFS.
References [1] C. W. Chou, D. B. Hume, J. C. J. Koelemeij, D. J. Wineland, and T. Rosenband. Frequencycomparison of two high-accuracy Al + optical clocks. Phys. Rev. Lett. , 104:070802, 2010.[2] A. A. Madej, P. Dub´e, Z. Zhou, J. E. Bernard, and M. Gertsvolf. Sr + − level via control and cancellation of systematic uncertainties and itsmeasurement against the SI second. Phys. Rev. Lett. , 109:203002, 2012. east-squares analysis of clock frequency comparison data [3] R. Le Targat, L. Lorini, Y. Le Coq, M. Zawada, J. Gu´ena, M. Abgrall, M. Gurov, P. Rosenbusch,D. G. Rovera, B. Nag´oney, R. Gartman, P. G. Westergaard, M. E. Tobar, M. Lours,G. Santarelli, A. Clairon, S. Bize, P. Laurent, P. Lemonde, and J. Lodewyck. Experimentalrealization of an optical second with strontium lattice clocks. Nat. Commun. , 4:2109, 2013.[4] N. Hinkley, J. A. Sherman, N. B. Phillips, M. Schioppo, N. D. Lemke, K. Beloy, M. Pizzocaro,C. W. Oates, and A. D. Ludlow. An atomic clock with 10 − instability. Science , 341:1215–1218, 2013.[5] B. J. Bloom, T. L. Nicholson, J. R. Williams, S. L. Campbell, M. Bishof, X. Zhang, W. Zhang,S. L. Bromley, and J. Ye. An optical lattice clock with accuracy and stability at the 10 − level. Nature , 506:71–75, 2014.[6] G. P. Barwood, G. Huang, H. A. Klein, L. A. M. Johnson, S. A. King, H. S. Margolis,K. Szymaniec, and P. Gill. Agreement between two Sr + optical clocks to 4 parts in 10 . Phys. Rev. A , 89:050501, 2014.[7] S. Falke, N. Lemke, C. Grebing, B. Lipphardt, S. Weyers, V. Gerginov, N. Huntemann,C. Hagemann, A. A-Masoudi, S. H¨afner, S. Vogt, U. Sterr, and C. Lisdat. A strontiumlattice clock with 3 × − inaccuracy and its frequency. New. J. Phys. , 16:073023, 2014.[8] R. M. Godun, P. B. R. Nisbet-Jones, J. M. Jones, S. A. King, L. A. M. Johnson, H. S. Margolis,K. Szymaniec, S. N. Lea, K. Bongs, and P. Gill. Frequency ratio of two optical clocktransitions in Yb + and constraints on the time-variation of fundamental constants. Phys.Rev. Lett. , 113:210801, 2014.[9] N. Huntemann, B. Lipphardt, C. Tamm, V. Gerginov, S. Weyers, and E. Peik. Improved limiton a temporal variation of m p /m e from comparisons of Yb + and Cs atomic clocks. Phys.Rev. Lett. , 113:210802, 2014.[10] I. Ushijima, M. Takamoto, M. Das, T. Ohkubo, and H. Katori. Cryogenic optical lattice clockswith a relative frequency difference of 1 × − . arXiv:1405.4071 , 2014.[11] T. L. Nicholson, S. L. Campbell, R. B. Hutson, G. E. Marti, B. J. Bloom, R. L. McNally,W. Zhang, M. D. Barrett, M. S. Safronova, G. F. Strouse, W. L. Tew, and J. Ye. 2 × − uncertainty in an atomic clock. arXiv:1412.8261 , 2014.[12] F. Levi, D. Calonico, L. Lorini, and A. Godone. IEN-CsF1 primary frequency standard atINRIM: accuracy evaluation and TAI calibrations. Metrologia , 43:545–555, 2006.[13] S. Weyers, V. Gerginov, N. Nemitz, R. Li, and K. Gibble. Distributed cavity phase frequencyshifts of the caesium fountain PTB-CsF2.
Metrologia , 49:82–87, 2012.[14] J. Gu´ena, M. Abgrall, D. Rovera, P. Laurent, B. Chupin, M. Lours, G. Santarelli, P. Rosenbusch,M. E. Tobar, R. Li, K. Gibble, A. Clairon, and S. Bize. Progress in atomic fountains at LNE-SYRTE.
IEEE Trans. Ultrason. Ferroelectr. Freq. Control , 59:391–410, 2012.[15] K. Szymaniec, S. Lea, and K. Liu. An evaluation of the frequency shift caused by collisionswith background gas in the primary frequency standard NPL-CsF2.
IEEE Trans. Ultrason.Ferroelectr. Freq. Control , 61:203–206, 2014.[16] T. P. Heavner, E. A. Donley, F. Levi, G. Costanzo, T. E. Parker, J. H. Shirley, N. Ashby,S. Barlow, and S. R. Jefferts. First accuracy evaluation of NIST-F2.
Metrologia , 51:174–182,2014.[17] P. Gill. When should we change the definition of the second?
Proc. Roy. Soc. A + and Hg + single-ionoptical clocks; metrology at the 17th decimal place. Science , 319:1808–1811, 2008.[20] K. Matsubara, H. Hachisu, Y. Li, S. Nagano, C. Locke, A. Nogami, M. Kajita, K. Hayasaka,T. Ido, and M. Hosokawa. Direct comparison of a Ca + single-ion clock against a Sr latticeclock to verify the absolute frequency measurement. Opt. Express , 20:22034–22041, 2012.[21] D. Akamatsu, M. Yasuda, H. Inaba, K. Hosaka, T. Tanabe, A. Onae, and F.-L. Hong. Frequencyratio measurement of
Yb and Sr optical lattice clocks.
Opt. Express , 22:7898–7905, 2014.[22] H. S. Margolis, R. M. Godun, P. Gill, L. A. M. Johnson, S. L. Shemar, P. B. Whibberley,D. Calonico, F. Levi, L. Lorini, M. Pizzocaro, P. Delva, S. Bize, J. Achkar, H. Denker,L. Timmen, C. Voigt, S. Falke, D. Piester, C. Lisdat, U. Sterr, S. Vogt, S. Weyers,J. Gersl, T. Lindvall, and M. Merimaa. International timescales with optical clocks (ITOC).In
Proceedings of the 2013 Join European Frequency and Time Forum and InternationalFrequency Control Symposium , pages 908–911. IEEE, 2013.[23] P. J. Mohr and B. N. Taylor. CODATA recommended values of the fundamental physical east-squares analysis of clock frequency comparison data constants: 1998. Rev. Mod. Phys. , 72:351–495, 2000.[24] J. R. D’Errico. HPF class, 2012. Available from the Matlab central file exchange, .[25] A. Matveev, C. G. Parthey, K. Predehl, J. Alnis, A. Beyer, R. Holzwarth, T. Udem, T. Wilken,N. Kolachevsky, M. Abgrall, D. Rovera, C. Salomon, P. Laurent, G. Grosche, O. Terra,T. Legero, H. Schnatz, S. Weyers, B. Altschul, and T. W. H¨ansch. Precise measurement ofthe hydrogen 1S–2S frequency via a 920-km fiber link.
Phys. Rev. Lett. , 110:230801, 2013.[26] C. Tamm, N. Huntemann, B. Lipphardt, V. Gerginov, N. Nemitz, M. Kazda, S. Weyers, andE. Peik. Cs-based optical frequency measurement using cross-linked optical and microwaveoscillators.
Phys. Rev. A , 89:023820, 2014.[27] C. Y. Park, D.-H. Yu, W.-K. Lee, S. E. Park, E. B. Kim, S. K. Lee, J. W. Cho, T. H.Yoon, J. Mun, S. J. Park, T. Y. Kwon, and S.-B. Lee. Absolute frequency measurementof S ( F = 1 / P ( F = 1 /
2) transition of
Yb atoms in a one-dimensional optical latticeat KRISS.
Metrologia , 50:119–128, 2013.[28] D. Akamatsu, H. Inaba, K. Hosaka, M. Yasuda, A. Onae, T. Suzuyama, M. Amemiya, andF.-L. Hong. Spectroscopy and frequency measurement of the Sr clock transition by laserlinewidth transfer using an optical frequency comb.