Months-long real-time generation of a time scale based on an optical clock

Time scales consistently provide precise time stamps and time intervals by combining atomic frequency standards with a reliable local oscillator. Optical frequency standards, however, have not been applied to the generation of time scales, although they provide superb accuracy and stability these days. Here, by steering an oscillator frequency based on the intermittent operation of a 87Sr optical lattice clock, we realized an “optically steered” time scale TA(Sr) that was continuously generated for half a year. The resultant time scale was as stable as International Atomic Time (TAI) with its accuracy at the 10−16 level. We also compared the time scale with TT(BIPM16). TT(BIPM) is computed in deferred time each January based on a weighted average of the evaluations of the frequency of TAI using primary and secondary frequency standards. The variation of the time difference TA(Sr) – TT(BIPM16) was 0.79 ns after 5 months, suggesting the compatibility of using optical clocks for time scale generation. The steady signal also demonstrated the capability to evaluate one-month mean scale intervals of TAI over all six months with comparable uncertainties to those of primary frequency standards (PFSs).

could be to use an OFS directly as a signal source of UTC(k). However, a more realistic way being currently discussed is steering a local oscillator with reference to an OFS instead of a microwave frequency standard. The feasibility of this method with an OFS was investigated by post-processing using data of OFS operations over 10 − 80 days 11,12 . In particular, it was reported by Grebing et al. 12 that they made a simulation and predicted that the nearly continuous operation of an OFS in future may realize a time scale which is superior to those provided by a state-of-the-art PFS.
Here, in this article, we propose and demonstrate an OFS-based time scale, which only requires temporally sparse operation of an OFS. Together with the operation of a stable HM as a flywheel, an accurate 87 Sr lattice clock for the measurement of the maser frequency was operated for 10 4 s approximately once a week, and the frequency of the flywheel was continuously steered for half a year with reference to the calibrations, leading to a real-time signal of a stable time scale with accurate scale intervals. Our intermittent operations of the OFS also enabled the one-month mean of the TAI scale interval to be estimated uninterruptedly throughout the half year. Evaluations by using an optical clock with an up-time ratio below 2% make it much easier for OFSs to contribute to TAI, suggesting a novel approach to maintain the accuracy of the TAI scale interval. For this goal, we also discuss the requirements of the OFS operation rate and HM instability, which should be useful for applying the proposed scheme to other combinations of OFSs and flywheels.

Results
Architecture of the time scales steered by an intermittently available OFS. Regardless of the rapid progress in OFSs, the continuous operation of OFSs has not become an easy task owing to the difficulty in maintaining the whole complicated system at its optimum state. Thus, we need to investigate what level of continuous operation is required for an OFS to accurately steer the flywheel frequency. A clue to addressing this issue is found in Fig. 1, where the frequency instabilities of various oscillators including those used in this work are shown. HMs are commonly used as a flywheel owing to their excellent balance between short-and long-term frequency stabilities. The instabilities of HMs are normally lowest at around 10 4 -10 5 s in terms of the Allan deviation. This is due to the existence of a linear frequency drift, and such characteristics are found in Fig. 1 from the instabilities of HM1 and HM2 (red and blue) that we used in this work. The red curve shows that the HM1 frequency reaches an instability of around 3 × 10 −16 . To keep the stability at the 10 −16 level, the frequency of HM1 must be steered to an external stable frequency reference. Cs fountain frequency standards play the role of the calibrator owing to their stable and accurate frequency in the long term, although they require a long averaging time for high precision. In the case of conventional Cs fountains using an HM as a reference 13 , it takes more than a week to evaluate a frequency with a statistical uncertainty of 5 × 10 −16 (orange line in Fig. 1). Even for a more stable Cs fountain employing a cryogenic oscillator 14 or photonically generated microwave 15 as a reference (green line), operation for over 3 × 10 4 s is still necessary 16 .
The instability of OFSs, on the other hand, is much lower than that of microwave frequency standards. Since the instability of the Sr optical lattice clock employed here (purple) is an order of magnitude lower than that of other microwave standards, the instability in measuring the frequency of an HM using an OFS is determined solely by the HM. Overall, the plots in Fig. 1 indicate that the operation of an OFS for only 10 4 s is required to evaluate the HM frequency with accuracy at the mid-10 −16 level. This is much shorter than for the case of using Cs fountains. In longer time scale, the temporal variation of an HM frequency comprises a linear frequency drift and a stochastic phase fluctuation 17,18 . Thus, the prediction of the linear drift from the latest records of the HM frequency makes it possible to cancel the frequency drift to a large extent. Then, the residual noise, which is Time scale generation by steering a maser frequency with respect to an optical clock. Our system for the generation and evaluation of an optically steered time scale is depicted in Fig. 2. For the generation, an HM (HM1) and a 87 Sr lattice clock (NICT-Sr1) act as a flywheel and a calibrator, respectively. As shown in Fig. 1, HM1 has instability at the 4 × 10 −16 level at an averaging time of 1 × 10 4 s, from which the Hadamard deviation is extended to a flicker floor of around 3 × 10 −16 up to three weeks. Thus, we operate NICT-Sr1 for only 10 4 s per operation, after which the time scale relies on HM1 until the next OFS operation. NICT-Sr1 as a calibrator has a systematic uncertainty at the 10 −17 level 19 , which proves the validity of utilizing it as an accurate frequency reference. The steering was based on the absolute frequency, which we determined in Ref. 20 .
The offset and drift in HM1 frequency were corrected using a phase micro stepper (PMS). With reference to HM1, the PMS generates a frequency shifted from HM1 by a small amount. The linearly drifting HM1 frequency between two OFS operations was estimated from the time series of the HM1 frequencies evaluated by NICT-Sr1 in the past. Then, the corresponding frequency offset at the moment was repeatedly estimated and added to the PMS with the opposite sign, resulting in a drift-free signal, TA(Sr). Thus, TA(Sr) was generated independently without any reference time scales.
The free-running HM1 signal at 100 MHz was compared with the 100 MHz signal downconverted from NICT-Sr1 by recording the phase difference. The mean frequency of HM1 with respect to the OFS-based reference was calculated from the differential phase between the start and end of the OFS operation. Thus, it is convenient to introduce the fractional frequency of HM1 y i for the ith OFS operation as where f HM1 , f OFS , φ F , φ I , and Δ op are the HM1 frequency, the OFS-based reference frequency, the relative phases at the end and start, and the duration of the OFS operation, respectively. Note that φ F and φ I are in the unit of seconds, resulting in y i being a fractional quantity. On the basis of y i (i = 0…, N) obtained in the past time interval T, the linear drift of y(t) was predicted as shown in Fig. 3a. Here, t i and t S are the time of the ith operation of the OFS and the time when the prediction of Figure 2. Schematic diagram of the realization and evaluation of an optically steered time scale. A strontium lattice clock and a frequency comb stabilized to the clock laser generate a microwave with a frequency of exactly 100 MHz. We operate the lattice clock once a week or more frequently, and record the differential phase between HM1 and the OFS-based microwave at the start and end of the operation, from which the mean fractional frequency of HM1 is calculated. The results obtained in the past 25 days allow us to predict the linear drift of the HM1 frequency. A phase micro stepper (PMS) externally shifts the HM1 frequency slightly so as to cancel the predicted linear drift, resulting in the time scale TA(Sr). TA(Sr) and UTC are mediated by UTC(NICT). Thus, the accuracy and stability of TA(Sr) were evaluated by obtaining the time difference from the virtual "paper clocks" UTC and TT(BIPM16). DMTD: dual mixer time difference system 21 , JST: Japan Standard Time. the linear drift is updated, respectively. It is assumed here that the OFS was operated N + 1 times inside the drift evaluation interval of t S − T < t < t S . The latency for incorporating new measurement results is less than 18 h, which is negligible compared with other parameters such as T and t i + 1 − t i . At time t S which occurs every 4 h regardless of OFS operations, linear fitting to y i (i = 0, …, N;t S − T < t i < t S ) was performed, resulting in the estimation ŷ t ( ) of the HM fractional frequency y(t) beinĝ where y t ( ) S and d(t S ) are the offset and drift rate of the fitting predicted at t S , respectively. Note that ŷ t ( ) changes not only immediately after the OFS operation but also when the earliest evaluation y 0 becomes beyond the drift evaluation interval t S − T < t < t S . We need to consider a tradeoff in the determination of T. More data points in a longer T may reduce the statistical error in the linear fitting, whereas the fitting over a long duration suffers from the error due to the quadratic and higher order terms of y(t). We set T to 25 days, until which y(t) is well approximated as linear because the flicker floor of the Hadamard deviation σ H (τ) of HM1 remains for up to 20 days (Fig. 1). The OFS operation was performed once a week or more frequently to obtain four or more points for the Figure 3. Scheme of frequency steering. a, As shown by the red dashed line, the HM frequency y(t) was predicted by extrapolation of a linear fitting to past HM frequencies y i (i = 0…N), which were obtained in the latest duration of T. Here, the case of N = 5 is drawn. The steering frequency Δy PMS (t) was changed every 4 h. We determined the value of Δy PMS (t) entered in the PMS so that the time integration of + Δ y t y t ( ) ( ) PMŜ becomes null; in other words, −Δy PMS (t) and y t ( ) intersect at the middle of one step. Note that the number of 4 h intervals between y i and y i+1 shown here is much smaller than that of the real situation since two calibrations are typically separated for one week. b, Change in the frequency of a hydrogen maser (HM1) evaluated using the fitting. This corresponds to N = 3 or more. Soon after y t ( ) is renewed at t S , we renewed the steering frequency at PMS Δy PMS (t), to cancel ŷ t ( ). Considering the potential noise in changing Δy PMS (t), the update rate of Δy PMS (t) was suppressed to once every 4 h.
We Comparison with reference time scales. For the characterization of TA(Sr), we need other independent reference time scales for comparison. We chose UTC and TT(BIPM) as references since they are linked to UTC(NICT). UTC(NICT) is an atomic time scale that is used as the origin of Japan Standard Time (JST). The phases of HM1, HM2, and TA(Sr) with respect to UTC(NICT) are continuously monitored by a precise phase measurement system (DMTD, dual mixer time difference system) 21 contained in the JST system 22 . The time difference of UTC(NICT) relative to other UTC(k) is routinely measured by satellite-based precision time and frequency transfer methods 1 . By incorporating such data sent from public laboratories worldwide, BIPM publishes UTC(NICT)-UTC and its uncertainty in the monthly report Circular T. The blue points in Fig. 4 show the time difference TA(Sr) − UTC. While TA(Sr) is a real-time signal, UTC is virtual and available only every 5 days. The time offset was initialized to zero on May 1st, 2016 (MJD 57509). TA(Sr) gradually became delayed relative to UTC and the difference reached 8.3 ns after five months. This monotonic change corresponds to a mean frequency bias of 6 × 10 −16 . This, however, does not indicate that the scale interval of TA(Sr) is different from that of the SI second by that amount. Instead, this deviation is caused by the fact that the scale interval of UTC was smaller than that of the SI second. The magnitude of the deviation is suggested in Circular T, where the scale interval of UTC is estimated by considering the calibration using the results reported from PFSs worldwide. Circular T Nos. 341-345 indicate that the scale interval of UTC was shorter than the SI second by a fractional amount at the mid-10 −16 level during our study.
We can discuss this point quantitatively by comparison with the time scale, TT(BIPMXY) 23 . The scale interval of TT(BIPMXY) is highly accurate since it is computed annually by a postprocessing that incorporates all calibrations provided by primary or secondary frequency standards worldwide. The "XY" denotes the last year of the computation interval. The most recent time scale, TT(BIPM16), was issued in January 2017, and the relative difference TT(BIPM16) -UTC is drawn as a black dashed line in Fig. 4 with the offset nulled on May 1st. It is clear that TA(Sr) shows good agreement with TT(BIPM16). The difference TA(Sr) -TT(BIPM16) is drawn as red points and was 0.79 ns after five months' generation.
The superb stability of TA(Sr) is observed as the overlapping Allan deviation of TA(Sr) -UTC shown in the inset. Note that this instability includes the fluctuation of the time link between UTC and UTC(NICT). This extra uncertainty is hidden in the instability between TA(Sr) and UTC. As we discussed in Ref. 19 , the link instability appears for up to an averaging time of 10 days. For an averaging time of 20 days, however, the link uncertainty is reduced to 2 × 10 −16 , and the instability of 3.9 × 10 −16 at 20 days shown in the inset is predominantly determined by the instability of TA(Sr) and UTC. Considering that UTC has an instability of 3 × 10 −16 at one month 1 , the residual instability of 1.5 × 10 −16 could be attributed to the instability of TA(Sr), demonstrating the instability of TA(Sr) that rivals state-of-the-art paper clocks.

Requirement for the HM stability and OFS operation rate.
For the time scale steered by the intermittent operation of an OFS, it is valuable to investigate how often we need to operate optical clocks depending on the stability of the flywheel. The uncertainty of the time scale is predominantly attributed to the prediction error of the linear trend as well as the stochastic phase of the flywheel. It is possible to calculate these uncertainties analytically by considering the error of least-squares fitting as well as the noise characteristics of the flywheel. The estimation, whose details are described in "Methods", results in the deviation of the phase accumulated between two OFS operations being where it is assumed that N + 1 operations are homogeneously distributed in T, σ p is the statistical uncertainty in the evaluation of the flywheel frequency by OFS operation, and σ F is the magnitude of the flicker floor in the Hadamard deviation of the flywheel. The contributions originating from σ p and σ F to the total phase error are denoted as ε p and ε F , respectively. The dependence of E[|Δφ|] on N is shown in Fig. 5 as black squares along with ε p and ε F . Here, the parameters σ p and σ F are 4 × 10 −16 and 3 × 10 −16 , respectively, and we consider the case of T = 30 days for generality. Thus, N corresponds to the number of calibrations per month. E[|Δφ|]N 1/2 , namely the red points in Fig. 5, is the integrated extra phase in one month for the ideal case that Δφ is uncorrelated with the adjacent free evolution. This hypothesis is introduced as a simplification, although the detrended HM shows not white frequency noise but flicker frequency noise. Nevertheless, we can analytically estimate the possible phase error owing to this simplification, and we consider that E[|Δφ|]N 1/2 can still be used as a guide to choose the parameters relevant to the intermittent optical steering. The estimation of the phase error integrated over five months, namely E[|Δφ|] (5N) 1/2 , is also shown as blue points to compare with our result. Here, the case of N = 4, namely, OFS operations once a week, yields a time difference of 1.5 ns. This level is similar to the final difference of 0.79 ns obtained in our experiment as shown in Fig. 4. Figure 5 also shows that the extra phase in one month is 0.66 ns in the case of the once-a-week operation. Taking into account the fact that the notification of UTC by Circular T has a mean latency of one month, it seems that maintaining the synchronization of UTC(k) within 1 ns is feasible for the optically steered time scale.
Simulation for the case of another noisier HM and infrequent OFS operation. The dependences of the time scale on parameters such as HM stability and OFS operation rate were investigated by simulations based on the actual record of the phase in HM1 as well as another HM (HM2) with a higher flicker floor level of 5 × 10 −16 . The JST system contains both HM1 and HM2, and records their relative phase with respect to UTC(NICT) every second. Using this record, we can extract the mean frequency ratio of the two HMs over an  Fig. 3b is converted to the value for HM2, with which we can determine the adjustment frequency at a virtual PMS used for HM2, enabling the simulation of the case that HM2 is used as the flywheel.
Furthermore, it is possible in simulations to partly omit the steering to investigate the dependence of the instability on the OFS operation rate. It is better in principle to perform OFS operations with a temporally homogeneous distribution as an HM has flicker noise in the Hadamard deviation. All points in a lumped evaluation may lie on one side of the linear trend, which may result in the incorrect prediction of the linear frequency drift. Considering these issues, we picked the OFS operations that realize a rather homogeneous distribution of once a week or once every two weeks, and made simulations for the cases of HM1 and HM2 as the flywheel oscillator. These cases correspond to the parameters of T = 21 days and N = 3 for the once-a-week calibration, and T = 28 days, N = 2 for the calibration every two weeks. Note that it is possible to obtain two curves for the case of evaluations once every two weeks by choosing odd and even weeks.
The results of six simulations (HM1 and HM2 every week and on odd and even weeks) were compared with that of TT(BIPM16) as shown in Fig. 6, where the expected phase excursions derived from the simple estimation discussed above are also shown as shaded areas. Here, we ignored the UTC-UTC(NICT) link error since it is negligibly small for an averaging time of more than 20 days. It is clear that the time scales using HM1 (solid curves) are more stable than those using HM2 (dashed curves), reflecting the difference in the flicker floor. The cases of infrequent evaluations once every two weeks are drawn as green and orange curves, denoting the choice of odd and even weeks, respectively. Having the half rate of steering causes larger phase excursions for the case of HM2. We can derive the instabilities from the simulation results shown in Fig. 6. The overlapping Allan deviation over 40 days is around 3 × 10 −16 for the two operation rates using HM1. The deviations using HM2 with the calibration rates of once a week and once every two weeks were 5 and 7 parts per 10 16 , respectively.
Evaluation of the TAI scale interval. Once a highly stable time scale is realized, the mean frequency difference between TAI and TA(Sr) in a longer interval is derived using equation (1). Employing a longer interval is critically important to reduce the uncertainty as TAI-TA(Sr) always includes the uncertainty of the time link UTC-UTC(NICT). The one-month mean frequency of UTC is normally calibrated using primary or secondary frequency standards, and the result is published in Circular T as the fractional deviation of the scale interval of TAI from that of the SI second.
The resultant mean frequency difference between TAI and TA(Sr) is shown in Fig. 7 as red open circles, where the difference was converted to a fractional scale interval following the notation in Circular T. It also shows the results of the calibration using other references such as TT(BIPM16) and state-of-the-art PFSs (PTB-CSF2 24 and SYRTE-FO2 14 ); the estimation of the uncertainty for the result of TA(Sr) is not straightforward since the uncertainty depends on the number of OFS operations as well as on how homogeneously the measurements are distributed in time 18,25 .
We therefore employed another method of TAI calibration that allows an easier estimation of the uncertainty. This method does not use all the results of OFS operations but selects five or six data for the month that are separated by nearly one week, and the linear drift in the fractional mean frequency of the HM is estimated by linear fitting using those five or six points. We can evaluate the uncertainty in this case by assuming that the intermittent measurements are homogeneously distributed over one month. Furthermore, as a real-time signal is not required for the TAI calibration, it is possible to use the mean phase of multiple HMs to mitigate the effect of a sporadic phase excursion of a specific HM 19 . Table 1 shows a typical uncertainty budget in this proposed scheme of the TAI calibration. The uncertainties due to the extra phase of HM in the OFS down time are dominant contributors. The details of the contributions are described in "Methods". The total uncertainty without the uncertainty of standard frequency amounted to 3.6 × 10 −16 , which was at a level similar to those in the evaluation using the state-of-the-art PFS. The absolute frequency of the 87 Sr clock transition has been measured in various laboratories over the years, where details including references are given in "Methods". The weighted mean of all absolute Here, we adopted this ν mean as the standard frequency, which leads to the total uncertainty of the TAI calibration being 3.7 × 10 −16 .
The result of this TAI calibration using NICT-Sr1 combined with the mean of two HMs (HM1 and HM2) is depicted in Fig. 7 as red filled circles. Here, we adopted the mean as the standard frequency. It shows that NICT-Sr1 was able to derive the scale interval of TAI over the half year. The results of NICT-Sr1 are consistent with the calibrations provided by the other two PFSs. This consistency is quantitatively investigated as follows. First, we calculated the mean of the three calibration results provided by the two Cs fountains and NICT-Sr1. Then, we obtained the mean deviations of the calibrations from this mean for the three references, which were 6.3 × 10 −16 , 7.8 × 10 −16 and 2.6 × 10 −16 for PTB-CSF2, SYRTE-FO2, and NICT-Sr1, respectively. The deviation of the NICT-Sr1 calibration is smaller than those of the two PFSs, indicating that the evaluation using Sr is consistent with those using other PFSs. In addition, the deviations of the calibrations by PTB-CSF2, SYRTE-FO2, and NICT-Sr1 with respect to the calibration by TT(BIPM16) were also calculated to be 2.3 × 10 −16 , 2.8 × 10 −16 , and 2.8 × 10 −16 , respectively. Note that TT(BIPM16) is effectively the average of all calibrations by PFSs, among which PTB-CSF2 and SYRTE-FO2 are the main contributors, and consequently, TT(BIPM16) and the two fountains have strong correlations. In spite of the correlations, the calibration by NICT-Sr1 is not significantly different from that by TT(BIPM16). These two investigations demonstrated that the intermittent operation of an OFS can be used for calibrating TAI with a similar level of uncertainty to that regularly provided by PFSs.

Discussion
State-of-the-art Cs fountains have realized accuracy below 10 −15 . While some fountains 14,24 including the Rb fountains in SYRTE 26   Standard frequency (u SDOM ) 9 Total 37 Table 1. Uncertainty budgets in the evaluation of the TAI scale interval using the intermittent operation of an OFS. All numbers are in parts of 10 −17 . Details of the contributors are described in "Methods". their operations. The community of time and frequency standards is instead inclined to develop optical standards. This situation is reflected in the calibration of the TAI scale interval by contributing PFSs, which are reported in Circular T. Smaller number of calibrations have been reported in the last few years, implying the reduced redundancy of PFSs in maintaining TAI at the low 10 −16 level. Ideally, the greatly improved accuracy and stability of OFSs would be reflected in the calibration of TAI. However, the achievable accuracy for TAI calibration is currently limited by the link uncertainty due to the satellite-based time comparisons between a local clock and TAI, requiring a long averaging time. Considering these circumstances, it is productive to generate time scales using the intermittent operation of optical clocks as demonstrated here until improved links, such as intercontinental fiber links, or satellite-based optical frequency comparisons 28 allow TAI to fully benefit from the superb accuracy and stability of OFSs distributed worldwide. The local oscillator in such a case may also be replaced with an optical oscillator using an optical cavity with low long-term instability. Monocrystalline silicon 29 is a promising material for such an optical oscillator as it has intrinsically low aging drift. The combination of fiber link networks and OFSs will further increase the reliability of optically steered time scales because various optical standards in separate laboratories linked by fibers can share the task of providing frequent calibrations for the steering.

Methods
Sr lattice clock. The details of NICT-Sr1, the 87 Sr lattice clock operated in this work, are described in Ref. 18,19 .
The Sr atoms are laser-cooled using a two-stage laser cooling technique and loaded to a vertically oriented one-dimensional optical lattice. The total systematic uncertainty of the Sr system was reduced over the course of six months from 8.4 to 5.0 parts per 10 17 predominantly by eliminating stray electric charge on the chamber windows 19 . Thus, the systematic uncertainty owing to the OFS is always less than that of the most accurate PFS 30 . The major systematic uncertainties in our system are the blackbody radiation shift and lattice light shift. We employed the absolute frequency obtained in Ref. 20 for the steering, which was 5 × 10 −16 lower than that of the CIPM recommendation on that date. This frequency is consistent with those evaluated in other laboratories 12,31 with a fractional difference of less than 2 × 10 −16 .
The optical frequency at the wavelength of 698 nm as the output of NICT-Sr1 is first downconverted to a microwave frequency using a commercial Er-fiber frequency comb as shown in Fig. 2. The downconverted signal with a frequency of exactly 1 GHz is derived from the fourth harmonic of the repetition rate (=250 MHz), assuming the absolute frequency of the clock transition obtained in Ref. 20 as described above. Then, the frequency of the 1 GHz microwave is divided tenfold to 100 MHz, and finally, the relative phase difference of the 100 MHz signal of HM1 from this optically generated microwave was recorded every second by a commercial frequency stability measurement set. The two 100 MHz signals are analog-to-digital converted in the instrument with a sampling rate of 64 Ms/s, from which the phase difference is recorded every second. The cumulative change in the phase difference during the measurement leads to a mean fractional frequency difference between the HM and NICT-Sr1. The phase measurement set has a system noise below 1 × 10 −17 over an integration time of 10 4 s.
Microwave system used to generate time scale and its link to TAI. The steering of the frequency is implemented using a PMS (Microsemi AOG-110). On the basis of an input signal at 5 MHz provided from HM1, the PMS can generate a signal with a fractional frequency shift of up to 5 × 10 −8 . The frequency offset is set with a resolution of 1 × 10 −19 , although the instability caused by the system noise of the PMS is 3 × 10 −13 /τ where τ is the averaging time in seconds. The 5 MHz signal steered by the Sr clock, namely TA(Sr), is sent to the dual mixer timing difference (DMTD) system contained in the JST system, where the phase difference between TA(Sr) and UTC(NICT) is recorded every second. The system noise of the DMTD system was measured to be 1 × 10 −16 with an averaging time of 10 4 s 21 . The GPS data are routinely obtained by feeding UTC(NICT) to a GPS receiver as a reference, by which the difference between UTC(NICT) and GPS time is obtained. This data is sent to BIPM together with time differences between UTC(NICT) and other local atomic clocks such as commercial Cs clocks and HMs. BIPM calculates a weighted mean of atomic clocks (Échelle Atomique Libre, EAL) 2 by analyzing such data from various institutes worldwide. Furthermore, BIPM compensates the frequency offset of EAL by incorporating the calibration provided by PFSs operated in NMIs. This correction was constant for more than four years until the end of 2016, which includes the period of the demonstration here. Thus, the deviation of the TAI scale interval relative to the SI second, which amounted to the 10 −16 level as shown in the past Circular T, was left as it was. This deviation is the object calibrated by PFSs and an optical lattice clock here.
BIPM computes another paper clock in annual postprocessing with an accuracy of around 2 parts in 10 16 by employing an algorithm to evaluate the scale interval of TAI with respect to calibration data randomly provided by PFSs 32  Estimation of the error due to the intermittent operation of the OFS. The performance of a time scale generated by intermittent optical steering is affected by the prediction error of the HM frequency. We estimated the magnitude of the error from the following analysis by using a simple noise model. The predictable trend of the HM frequency is expressed by the offset and the linear drift rate d of the frequency, whereas the residual fluctuation is characterized by the Hadamard deviation. Here, we employ a model in which the prediction errors of the HM phase between two temporally separated OFS operations comprise two factors, which are the prediction error of the linear trend of the frequency and the stochastically accumulated phase error 17 . In the following estimation, we set the conditions that the HM frequency is regularly calibrated with a temporal separation of ΔT and that N + 1 samples with fractional frequency y i (i = 0 … N) are obtained in the latest duration T = NΔT for drift estimation. We consider that the stochastic noise and residual white frequency noise cause a deviation Δy i from the linear trend. The dominant noises of the OFS operation for 10 4 s are flicker frequency noise and white frequency noise. The least-squares fitting to Δy i as a function of event time τ i = iΔT − T/2 gives a prediction error δd of the linear drift rate of The error of the offset in the linear regression is the mean of Δ = Δ y y ( ) i as τ i is distributed symmetrically around t = 0, which turns out to be Δ = . Thus, the phase error Δφ p in the next free-running due to the prediction error is written as Here, we assume the normal distribution of Δy i with a standard deviation of σ p . Then, the accumulated phase error ε p due to the prediction error of the linear trend in ΔT is calculated as where E[(Δφ p ) 2 ] is the expectation of (Δφ p ) 2 . The other factor is the accumulated stochastic phase error ε F in the duration T/2 < t < T/2 + ΔT, which is 33 in the case that the flicker frequency noise σ H (τ) = σ F is predominant. Thus, E[|Δφ|], the expected phase error between two OFS operations, is Uncertainty in TAI calibration. The total uncertainty in the estimation of the one-month mean scale interval of TAI, which is indicated as error bars in Fig. 7, was determined by considering the following contributions. They are classified into factors attributed to (i) the atomic frequency standard, (ii) the link between the frequency standard and TAI, and (iii) the standard frequency as a secondary representation of the second. The small up-time ratio here imposes a link uncertainty larger than the uncertainty owing to the atomic frequency standard. The details of the contributions are described as follows.
Atomic frequency standard. The uncertainty attributed to the Sr system comprises the type-A uncertainty u a and the type-B uncertainty u b . u a corresponds to the statistical uncertainty, which will be smaller if the measurement is performed for a longer time, whereas u b corresponds to the systematic uncertainty of the standard. For the TAI evaluation of the one-month mean, the one-month campaign includes five or six operations with 10 4 s OFS operations each. Since we evaluated the instability of NICT-Sr1 to be 7 × 10 −15 /τ 1/2 by comparing two alternative servos 34 , the estimated u a for the total interrogation time of τ = 5 × 10 4 s is 3.1 × 10 −17 . The type-B uncertainty, in other words, the systematic uncertainty of NICT-Sr1 slightly varied during the campaign as described above.
The mean with weights proportional to operation time was 6.9 × 10 −17 . Note that this uncertainty includes the uncertainty of the gravity shift, which amounts to 2.2 × 10 −17 .
Link. The time-link uncertainty comprises that in the laboratory u l/Lab and that for the satellite link u l/Tai following the notation adopted in Circular T. In the intermittent evaluation scheme proposed here, a major part of u l/Lab is the possible uncertainty in the estimation of the one-month mean frequency of the HM based on the limited number of OFS operations. The deviation of the HM frequency from the linear trend is caused by the stochastic phase fluctuation and residual white noise of the HM. The HM frequency after the removal of the linear drift shows a flicker noise floor for an averaging time of one day to three weeks as shown in Fig. 1. We performed five or six homogeneously distributed measurements with roughly one-week separation for the linear fitting, in which the one-month mean frequency is obtained as the intercept in the fitting. The expected error of the linear fitting based on the five or six (=N + 1) OFS calibrations is derived as follows. As discussed above, it is assumed that the measurements of the HM frequency deviate from the linear trend by Δy i . Then, using the error of the resultant linear drift d t y δ ⋅ + Δ , the cumulative phase error in −T/2 < t < T/2 due to the fitting error is Therefore, assuming a normal distribution of Δy i with a standard deviation of σ p = 4 × 10 −16 , the standard deviation of δΦ turns out to be σ p T/(N + 1) 1/2 , which corresponds to a one-month mean frequency difference of σ p /(N + 1) 1/2 = 1.8 × 10 −16 in the case of N = 4. The normal distribution of Δy i assumed here probably underestimates the error as flicker noise is dominant. It is worth comparing this estimated uncertainty with the standard error of the linear fitting performed to determine the predictable drift of the HM frequency. The standard error of the fitting was typically 2.6 × 10 −16 , which is clearly larger than σ p /(N + 1) 1/2 . Considering these issues, the uncertainty u l/HMtrend in the estimation of the linear drift is conservatively determined to be 2.6 × 10 −16 . In terms of the effects of the stochastic phase fluctuation of the HM, the induced phase noise in a dead time of τ is known to be τσ H /(ln2) 1/2 in the case of flicker frequency noise 33 . Thus, a non-operation time of one week results in a phase noise of 0.22 ns with σ H = 3 × 10 −16 . Since 30/7 periods of 7 days are contained in one month, the uncertainty of the stochastic part u l/HMstoch in the one-month mean frequency is estimated to be 0.22 × 10 −9 × 30/7/(86400 × 30) = 1.8 × 10 −16 .
Uncertainty in standard frequency. The calibration of frequency with respect to a reference transition provided by non-Cs atoms has an additional uncertainty owing to the absolute frequency of the reference transition. For the case of the 87 Sr clock transition, five absolute frequency measurements 12,19,20,31 were reported after the CIPM recommended frequency was updated in 2016 36 . All these fifteen results give a weighted mean frequency of ν mean = 429 228 004 229 873.04(4) Hz with the fractional standard deviation of the mean being u SDOM = 9.1 × 10 −17 . Note that the scattering of the data is presumably determined by the reproducibility of the SI second and frequency links.
Finally, the uncertainty without u SDOM , namely = + + + u u u u u ( ) sub a 2 b 2 l/Lab 2 l/Tai 2 1/2 , is 3.6 × 10 −16 . This uncertainty is at the same level as those obtained by the state-of-the-art PFSs. By incorporating the uncertainty of the standard frequency u SDOM , the total uncertainty for the TAI calibration proposed here amounts to 3.7 × 10 −16 .
Data availability. The datasets generated during and/or analysed during the current study are available from the corresponding author on reasonable request.