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

Hidekazu Hachisu  National Institute of Information and Communications
Technology, 4-2-1 Nukui-kitamachi, Koganei, Tokyo 184-8795, Japan  Fumimaru
Nakagawa  National Institute of Information and Communications Technology,
4-2-1 Nukui-kitamachi, Koganei, Tokyo 184-8795, Japan  Yuko Hanado  National
Institute of Information and Communications Technology, 4-2-1 Nukui-kitamachi,
Koganei, Tokyo 184-8795, Japan  Tetsuya Ido  National Institute of Information
and Communications Technology, 4-2-1 Nukui-kitamachi, Koganei, Tokyo 184-8795,
Japan<EMAIL_ADDRESS>
###### Abstract

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).

## Introduction

Time scales form a foundation of modern society, enabling technologies such as
global navigation satellite systems, radio-wave interferometer arrays, and the
synchronization of grids for telecommunication networks and electric power
systems. In general, stable frequency standards become an infrastructure when
signals are integrated to a time scale, by which the benefit of atomic
frequency standards is always available as timestamps or scale intervals.
While various time scales have been coordinated and operated[1], most national
standard times are adjusted to Coordinated Universal Time (UTC) since the
availability of UTC is solid due to the sufficient redundancy of the atomic
clocks involved. The International Bureau of Weights and Measures (BIPM)
receives information regarding more than 400 atomic clocks operated in
national metrology institutes and other public laboratories, and then
subsequently computes their weighted average[2, 3]. The frequency offset of
this mean free time scale with respect to the SI second is estimated using
primary frequency standards (PFSs) or secondary frequency standards. This
calibrated atomic mean time scale is called International Atomic Time (TAI).
UTC is the product after adding a phase step of integer seconds to TAI to
synchronize the time scale to Earth’s rotation [4]. It is noteworthy that UTC
is one of the so-called paper clocks, meaning a virtual product computed with
a latency of 15–40 days and that the accuracy of its scale interval relative
to the SI second lies at the $10^{-16}$ level. Furthermore, the computation
results are provided at limited instances at 0 h UTC on every fifth day. BIPM
also computes UTCr, which is computed every day and made available every week.
The sparse availability of UTC with latency requires national time and
frequency laboratories to generate their own time scales in real time for
their national standard time. Such a local time scale adjusted to UTC is
called UTC($k$), where $k$ stands for laboratory $k$ [1]. There are various
recipes for the generation of UTC($k$). The most stable one is currently
realized by steering the frequency of a hydrogen maser (HM) with respect to a
local Cs (or Rb) fountain standard [5, 6, 7, 8].

Microwave frequency standards, which presently realize all atomic time scales,
have made steady progress for more than fifty years since the SI second was
defined by the Cs hyperfine transition. However, advances in optical frequency
standards (OFSs) have accelerated in the last twenty years, resulting in
greatly improved stability and accuracy over microwave frequency standards
[9]. This situation has led the community of time and frequency standards to
seriously consider the future redefinition of the SI second, where the SI
second is realized by OFSs and the accuracy of the scale interval in TAI is
maintained by OFSs [10].

This trend to introduce OFSs may also activate attempts to realize more stable
UTC(k) with more accurate scale intervals. UTC($k$) needs to be a real-time
signal without interruptions. The simplest way to achieve this 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 ${}^{87}{\rm Sr}$ 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 $3\times 10^{-16}$. 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
$5\times 10^{-16}$ (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 $3\times 10^{4}$ 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
expressed by the three-sample variance (Hadamard variance) $\sigma_{\rm
H}^{2}\left(\tau\right)$, forms a flicker noise floor that extends to rather
long averaging time. This extended flicker floor allows the time scale to rely
on the HM for a certain amount of time. OFSs are highly advantageous in this
regard because their high stability enables the evaluation of the HM frequency
in a short averaging time. We examined the feasibility of this strategy in our
case considering the instability of available HMs. The Hadamard deviations of
${\rm HM1-HM2}$ divided by $\sqrt{2}$, UTC-HM1, and UTC-HM2 are shown in Fig.
1 as dashed curves, indicating flicker floors of HM1 and HM2 at the
$(3-5)\times 10^{-16}$ extending to an averaging time of one week or more.
Thus, it is feasible to compose a time scale with instability at the
$10^{-16}$ level by the intermittent operation of an OFS once a week.

Figure 1: Frequency instabilities of various oscillators, including microwave
and optical frequency standards as well as the two HMs and Sr lattice clock
used in this work. The much lower instability of the lattice clock allows the
HM mean frequencies to be evaluated in a shorter time. The dashed curves are
relevant overlapping Hadamard deviations. The deviation of HM1 vs HM2 divided
by $\sqrt{2}$ is shown as black squares for short averaging times of less than
10 days. The deviations of HM1 and HM2 for longer averaging times are
calculated against UTC.

### 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
${}^{87}{\rm Sr}$ lattice clock (NICT-Sr1) act as a flywheel and a calibrator,
respectively. As shown in Fig. 1, HM1 has instability at the $4\times
10^{-16}$ level at an averaging time of $1\times 10^{4}$ s, from which the
Hadamard deviation is extended to a flicker floor of around $3\times 10^{-16}$
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.

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 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
$i$th OFS operation as

$y_{i}=\frac{f_{\rm HM1}}{f_{\rm OFS}}-1=\frac{\phi_{\rm F}-\phi_{\rm
I}}{\Delta_{\rm op}},$ (1)

where $f_{\rm HM1},f_{\rm OFS},\phi_{\rm F},\phi_{\rm I},$ and $\Delta_{\rm
op}$ 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 $\phi_{\rm F}$ and $\phi_{\rm I}$ are in the unit of
seconds, resulting in $y_{i}$ being a fractional quantity.

On the basis of $y_{i}(i=0,\ldots,N)$ obtained in the past time interval $T$,
the linear drift of $y(t)$ was predicted as shown in Fig. 3(a). Here, $t_{i}$
and $t_{\rm S}$ are the time of the $i$th operation of the OFS and the time
when the prediction of the linear drift is updated, respectively. It is
assumed here that the OFS was operated $N+1$ times inside the drift evaluation
interval of $t_{\rm S}-T<t<t_{\rm S}$. 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 $t_{\rm S}$ which occurs
every 4 h regardless of OFS operations, linear fitting to
$y_{i}\left(i=0,\ldots,N;t_{\rm S}-T<t_{i}<t_{\rm S}\right)$ was performed,
resulting in the estimation $\hat{y}\left(t\right)$ of the HM fractional
frequency $y(t)$ being

$\hat{y}\left(t\right)=\overline{y(t_{\rm S})}+d\left(t_{\rm S}\right)\cdot
t,$ (2)

where $\overline{y(t_{\rm S})}$ and $d(t_{\rm S})$ are the offset and drift
rate of the fitting predicted at $t_{\rm S}$, respectively. Note that
$\hat{y}(t)$ changes not only immediately after the OFS operation but also
when the earliest evaluation $y_{0}$ becomes beyond the drift evaluation
interval $t_{\rm S}-T<t<t_{\rm S}$. 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 $\sigma_{\rm
H}(\tau)$ 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
fitting. This corresponds to $N=3$ or more. Soon after $\hat{y}(t)$ is renewed
at $t_{\rm S}$, we renewed the steering frequency at PMS $\Delta y_{\rm
PMS}(t)$, to cancel $\hat{y}(t)$. Considering the potential noise in changing
$\Delta y_{\rm PMS}(t)$, the update rate of $\Delta y_{\rm PMS}(t)$ was
suppressed to once every 4 h.

We began the intermittent operation of the OFS (NICT-Sr1) in April 2016. After
one month of preliminary operations for the initial characterization of HM1,
we started the steering on May 1st, 2016 (MJD 57509), and continued it for
five months to the end of September (MJD 57661). The red points in Fig. 3b
show the measured fractional frequency $y_{i}$ of HM1 relative to NICT-Sr1
over half a year. The results clearly show the stable linear drift of the HM1
frequency. The details of the evaluation in Fig. 3(b) are shown in Fig. 3c,
where the differences from the linear fitting line (green dashed line) in
Fig.3b are shown. The purple line in Fig. 3c shows $-\Delta y_{\rm PMS}(t)$.
The purple line has a fine sawlike structure with a period of 4 h since the
frequency correction at the PMS (blue line in Fig. 3(a)) is constant for 4 h.
The deviations of the red points from the purple line, which are the
difference in the prediction $\hat{y}(t_{N})$ from $y_{N}$, are shown in the
lower panel of Fig. 3c. In most cases, they are lower than $1\times 10^{-15}$.

### 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 ${\rm TA(Sr)-UTC}$. 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 $6\times 10^{-16}$. 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.

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 $y_{i}(i=0\ldots N)$, which were obtained in the latest
duration of $T$. Here, the case of $N=5$ is drawn. The steering frequency
$\Delta y_{\rm PMS}(t)$ was changed every 4 h. We determined the value of
$\Delta y_{\rm PMS}(t)$ entered in the PMS so that the time integration of
$\hat{y}(t)+\Delta y_{\rm PMS}(t)$ becomes null; in other words, $-\Delta
y_{\rm PMS}(t)$ and $\hat{y}(t)$ 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 ${}^{87}\rm Sr$ lattice clock. Stable linear
drift was found. c, The red points in b are shown in the upper panel as the
difference from the linear fitting (green dashed line in b). The magnitude of
the frequency shift subtracted at the PMS is also shown in purple. The purple
curve changes soon after a new red point updates the prediction of the HM
frequency, or at the instants when the earliest red point falls off from the
drift evaluation interval. The lower panel shows the difference in
$\hat{y}(t_{N})$ from $y_{N}$.

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
$2\times 10^{-16}$, and the instability of $3.9\times 10^{-16}$ 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 $3\times 10^{-16}$ at one
month [1], the residual instability of $1.5\times 10^{-16}$ could be
attributed to the instability of TA(Sr), demonstrating the instability of
TA(Sr) that rivals state-of-the-art paper clocks.

Figure 4: Behaviors of the realized time scale TA(Sr) and TT(BIPM16) with
respect to UTC (blue filled circles and dashed line, respectively). TA(Sr) is
consistent with TT(BIPM16). The red filled circles show the difference ${\rm
TA(Sr)}-{\rm TT(BIPM16)}$, showing the agreement at the sub-ns level after
five months. The inset shows the corresponding overlapping Allan deviation of
${\rm TA(Sr)}-{\rm UTC}$ and ${\rm TA(Sr)}-{\rm TT(BIPM16)}$.

### 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

$E\left[\left|\Delta\phi\right|\right]=\left(\epsilon_{\rm
p}^{2}+\epsilon_{\rm
F}^{2}\right)^{1/2}=\frac{T}{N}\left[\frac{(2N+1)(2N+3)}{N(N+1)(N+2)}\sigma_{\rm
p}^{2}+\frac{1}{\ln 2}\sigma_{\rm F}^{2}\right]^{1/2},$ (3)

where it is assumed that $N+1$ operations are homogeneously distributed in
$T$, $\sigma_{\rm p}$ is the statistical uncertainty in the evaluation of the
flywheel frequency by OFS operation, and $\sigma_{\rm F}$ is the magnitude of
the flicker floor in the Hadamard deviation of the flywheel. The contributions
originating from $\sigma_{\rm p}$ and $\sigma_{\rm F}$ to the total phase
error are denoted as $\epsilon_{\rm p}$ and $\epsilon_{\rm F}$, respectively.

The dependence of $E\left[|\Delta\phi|\right]$ on $N$ is shown in Fig. 5 as
black squares along with $\epsilon_{\rm p}$ and $\epsilon_{\rm F}$. Here, the
parameters $\sigma_{\rm p}$ and $\sigma_{\rm F}$ are $4\times 10^{-16}$ and
$3\times 10^{-16}$, respectively, and we consider the case of $T=30$ days for
generality. Thus, $N$ corresponds to the number of calibrations per month.
$E[|\Delta\phi|]N^{1/2}$, namely the red points in Fig. 5, is the integrated
extra phase in one month for the ideal case that $\Delta\phi$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
$E[|\Delta\phi|]N^{1/2}$ 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 $E[|\Delta\phi|](5N)^{1/2}$, 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.

Figure 5: Errors in an intermittently steered time scale as a function of the
number of OFS operations per month. The black filled squares are the errors
accumulated in one interval between two OFS operations. The error is
attributed to the fitting error in predicting the linear frequency drift (open
circles) as well as the stochastic phase of an HM (open triangles). The red
and blue points are estimations of the accumulated total error in one month
and five months, respectively. Note that these are analytically estimated
using equation (3).

### 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 $5\times 10^{-16}$. 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 OFS
operation time of $10^{4}$ s, enabling postprocessing evaluations of the HM2
frequency. Then, each point in 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 $3\times 10^{-16}$ 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.

Figure 6: Time differences from TT(BIPM16) simulated using actual record of HM
behaviors. The realized time scale TA(Sr) is also shown for reference as red
points. The blue solid curve is the simulation result when the flywheel
frequency was evaluated using the OFS almost regularly once a week. The green
and orange curves show the cases that the parameters were steered every two
weeks in odd and even weeks, respectively. The less stable HM, HM2, was
employed to obtain the three dashed curves.

### 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 $3.6\times 10^{-16}$, which
was at a level similar to those in the evaluation using the state-of-the-art
PFS. The absolute frequency of the ${}^{87}{\rm Sr}$ 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 frequency
measurements is $\nu_{\rm mean}=429\ 228\ 004\ 229\ 873.04$ Hz with a
fractional uncertainty of $9.1\times 10^{-17}$. Here, we adopted this
$\nu_{\rm mean}$ as the standard frequency, which leads to the total
uncertainty of the TAI calibration being $3.7\times 10^{-16}$.

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 $6.3\times 10^{-16},7.8\times
10^{-16}$, and $2.6\times 10^{-16}$ 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 $2.3\times 10^{-16},2.8\times 10^{-16}$,
and $2.8\times 10^{-16}$, 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.

Figure 7: Evaluation of one-month-mean TAI scale interval. The red open circles are calculated using the time scale TA(Sr), where the change of the time difference ${\rm UTC}-{\rm TA(Sr)}$in one month is divided by the duration of one month. The results obtained with the intermittent operation of the Sr lattice clock (red filled circles, see text) are consistent with those of other methods such as state-of-the-art PFSs (PTB-CSF2 and SYRTE-FO2). Sr atomic frequency standard |
---|---
statistical $(u_{\rm a})$ | 3
systematic $(u_{\rm b})$ | 7
Link |
$u_{\rm l/Lab}$ |
HM: trend estimation $(u_{\rm l/HMtrend})$ | 26
HM: stochastic phase noise $(u_{\rm l/HMstoch})$ | 18
DMTD system $(u_{\rm l/DMTD})$ | 5
$u_{\rm l/Tai}$ |
UTC-UTC(NICT) link | 14
Subtotal $(u_{\rm sub})$ | 36
Standard frequency $(u_{\rm SDOM})$ | 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”.

## 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] and USNO [27]
are operated almost continuously, fewer new fountains have recently begun
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 ${}^{87}{\rm Sr}$ 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
$5\times 10^{-16}$ 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 $2\times 10^{-16}$.

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 $1\times 10^{-17}$ 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 $5\times
10^{-8}$. The frequency offset is set with a resolution of $1\times 10^{-19}$,
although the instability caused by the system noise of the PMS is $3\times
10^{-13}/\tau$ where $\tau$ 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 $1\times 10^{-16}$ 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]. The resultant time scale is published as the time offset from TAI.
This time scale, TT(BIPM), is a realization of Terrestrial Time as defined by
the International Astronomical Union (IAU).

### 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 $\Delta T$ and
that $N+1$ samples with fractional frequency $y_{i}(i=0\ldots N)$ are obtained
in the latest duration $T=N\Delta T$ for drift estimation. We consider that
the stochastic noise and residual white frequency noise cause a deviation
$\Delta y_{i}$ 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 $\Delta y_{i}$ as a function of event time
$\tau_{i}=i\Delta T-T/2$ gives a prediction error $\delta d$ of the linear
drift rate of

$\delta d=\frac{12}{\left(\Delta
T\right)^{2}N(N+1)(N+2)}\sum^{N}_{i=0}\tau_{i}\Delta y_{i}.$ (4)

The error of the offset in the linear regression is the mean of $\Delta
y_{i}\left(=\overline{\Delta y}\right)$ as $\tau_{i}$ is distributed
symmetrically around $\tau=0$, which turns out to be $\overline{\Delta
y}=(N+1)^{-1}\sum^{N}_{i=0}\Delta y_{i}$. Thus, the phase error
$\Delta\phi_{\rm p}$ in the next free-running due to the prediction error is
written as

$\Delta\phi_{\rm p}=\int_{T/2}^{T/2+\Delta T}\left(\delta d\cdot
t+\overline{\Delta y}\right)dt=\frac{6}{N(N+2)}\sum^{N}_{i=0}\tau_{i}\Delta
y_{i}+\frac{T}{N(N+1)}\sum^{N}_{i=0}\Delta y_{i}.$ (5)

Here, we assume the normal distribution of $\Delta y_{i}$ with a standard
deviation of $\sigma_{\rm p}$. Then, the accumulated phase error
$\epsilon_{\rm p}$ due to the prediction error of the linear trend in $\Delta
T$ is calculated as

$\epsilon_{\rm p}\equiv\sqrt{E\left[\left(\Delta\phi_{\rm
p}\right)^{2}\right]}=\frac{T}{N}\left[\frac{(2N+1)(2N+3)}{N(N+1)(N+2)}\sigma_{\rm
p}^{2}\right]^{1/2},$ (6)

where $E\left[\left(\Delta\phi_{\rm p}\right)^{2}\right]$ is the expectation
of $\left(\Delta\phi_{\rm p}\right)^{2}$.

The other factor is the accumulated stochastic phase error $\epsilon_{F}$ in
the duration $T/2<t<T/2+\Delta T$, which is $\epsilon_{\rm
F}^{2}=\left(\sigma_{\rm F}\cdot\Delta T\right)^{2}/\ln 2\ $ [33] in the case
that the flicker frequency noise $\sigma_{\rm H}\textbf{}(\tau)=\sigma_{\rm
F}$ is predominant. Thus, $E\left[|\Delta\phi|\right]$, the expected phase
error between two OFS operations, is

$E\left[|\Delta\phi|\right]=\left(\epsilon_{\rm p}^{2}+\epsilon_{\rm
F}^{2}\right)^{1/2}=\frac{T}{N}\left[\frac{(2N+1)(2N+3)}{N(N+1)(N+2)}\sigma_{\rm
p}^{2}+\frac{1}{\ln 2}\sigma_{\rm F}^{2}\right]^{1/2}.$ (7)

### 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
$u_{\rm a}$ and the type-B uncertainty $u_{\rm b}$. $u_{\rm a}$ corresponds to
the statistical uncertainty, which will be smaller if the measurement is
performed for a longer time, whereas $u_{\rm b}$ 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 $7\times
10^{-15}/\tau^{1/2}$ by comparing two alternative servos [34], the estimated
$u_{\rm a}$ for the total interrogation time of $\tau=5\times 10^{4}$ s is
$3.1\times 10^{-17}$. 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 $6.9\times
10^{-17}$. Note that this uncertainty includes the uncertainty of the gravity
shift, which amounts to $2.2\times 10^{-17}$.

#### Link

The time-link uncertainty comprises that in the laboratory $u_{\rm l/Lab}$ and
that for the satellite link $u_{\rm l/Tai}$ following the notation adopted in
Circular T. In the intermittent evaluation scheme proposed here, a major part
of $u_{\rm l/Lab}$ 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 $\Delta
y_{i}$. Then, using the error of the resultant linear drift $\delta d\cdot
t+\overline{\Delta y}$, the cumulative phase error in $-T/2<t<T/2$ due to the
fitting error is

$\delta\Phi\left(|t|<T/2\right)=\int^{T/2}_{-T/2}\left(\delta d\cdot
t+\overline{\Delta y}\right)\ dt=\overline{\Delta
y}T=\frac{T}{N+1}\sum^{N}_{i=0}\Delta y_{i}.$ (8)

Therefore, assuming a normal distribution of $\Delta y_{i}$ with a standard
deviation of $\sigma_{\rm p}=4\times 10^{-16}$, the standard deviation of
$\delta\Phi$ turns out to be $\sigma_{\rm p}T/(N+1)^{1/2}$, which corresponds
to a one-month mean frequency difference of $\sigma_{\rm
p}/(N+1)^{1/2}=1.8\times 10^{-16}$ in the case of $N=4$. The normal
distribution of $\Delta y_{i}$ 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 $2.6\times 10^{-16}$, which is clearly larger than $\sigma_{\rm
p}/(N+1)^{1/2}$. Considering these issues, the uncertainty $u_{\rm l/HMtrend}$
in the estimation of the linear drift is conservatively determined to be
$2.6\times 10^{-16}$. In terms of the effects of the stochastic phase
fluctuation of the HM, the induced phase noise in a dead time of $\tau$ is
known to be $\tau\sigma_{\rm H}/(\ln 2)^{1/2}$ 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 $\sigma_{\rm H}=3\times 10^{-16}$. Since $30/7$
periods of 7 days are contained in one month, the uncertainty of the
stochastic part $u_{\rm l/HMstoch}$ in the one-month mean frequency is
estimated to be $0.22\times 10^{-9}\times\sqrt{30/7}/(86400\times
30)=1.8\times 10^{-16}$.

Another minor part of $u_{\rm l/Lab}$ is the uncertainty of the DMTD system
$u_{\rm l/DMTD}$. The uncertainty in five OFS operations amounts to $1\times
10^{-16}/\sqrt{5}=4.5\times 10^{-17}$ as the noise of the DMTD system is
$1\times 10^{-16}$ over an averaging time of $10^{4}$ s [21]. Finally, the
link uncertainty in a local laboratory $u_{\rm l/Lab}$ is $u_{\rm
l/Lab}=({u_{\rm l/HMtrend}}^{2}+{u_{\rm l/HMstoch}}^{2}+{u_{\rm
l/DMTD}}^{2})^{1/2}=3.2\times 10^{-16}$.

The UTC-UTC(NICT) link uncertainty of 0.21 ns which we estimated in Ref. 19
corresponds to the uncertainty in the 5-day mean frequency of $u_{\rm
5d}=(\sqrt{2}\times 0.21\times 10^{-9})/(5\times 86400)=6.9\times 10^{-16}$
[19]. For a longer averaging time $t_{\rm d}$ in days, it is known that the
uncertainty decreases to $u_{\rm 5d}(t_{\rm d}/5)^{-0.9}$ [35]. Thus, the
fractional uncertainty in the one-month mean frequency is $1.4\times
10^{-16}$.

#### 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 87Sr 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 $\nu_{\rm mean}=429\ 228\ 004\ 229\ 873.04(4)$ Hz
with the fractional standard deviation of the mean being $u_{\rm
SDOM}=9.1\times 10^{-17}$. Note that the scattering of the data is presumably
determined by the reproducibility of the SI second and frequency links.

Finally, the uncertainty without $u_{\rm SDOM}$, namely $u_{\rm sub}=({u_{\rm
a}}^{2}+{u_{\rm b}}^{2}+{u_{\rm l/Lab}}^{2}+{u_{\rm l/Tai}}^{2})^{1/2}$, is
$3.6\times 10^{-16}$. 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 $u_{\rm SDOM}$, the total uncertainty for the TAI calibration
proposed here amounts to $3.7\times 10^{-16}$.

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## Acknowledgements

The authors thank G. Petit, T. Gotoh, H. Ito, M. Kumagai and M. Hosokawa for
the fruitful discussions. Maintaining the operation of the lattice clock for
half a year was achieved with the technical help provided by H. Ishijima and
S. Ito.