# Roadmap towards the redefinition of the second

Noel Dimarcq, Marina Gertsvolf, Gaetano Mileti, Sebastien Bize, Christopher Oates, Ekkehard Peik, Davide Calonico, Tetsuya Ido, Patrizia Tavella, Frédéric Meynadier, et al.

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## To cite this version:

Noel Dimarcq, Marina Gertsvolf, Gaetano Mileti, Sebastien Bize, Christopher Oates, et al. Roadmap towards the redefinition of the second. Metrologia, 2023, 61 (1), pp.012001. 10.1088/1681-7575/ad17d2. hal-04375006v2

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# Roadmap towards the redefinition of the second

To cite this article: N Dimarcq *et al* 2024 *Metrologia* **61** 012001

View the [article online](#) for updates and enhancements.

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Review

Roadmap towards the redefinition of the
second

N Dimarcq<fnref n="1" />, M Gertsvolf<fnref n="2" /> ID, G Mileti<fnref n="3" />, S Bize<fnref n="4" /> ID, C W Oates<fnref n="5" />, E Peik<fnref n="6" />, D Calonico<fnref n="7" />,
T Ido<fnref n="8" /> ID, P Tavella<fnref n="9" />,* ID, F Meynadier<fnref n="9" /> ID, G Petit<fnref n="9" /> ID, G Panfilo<fnref n="9" />, J Bartholomew<fnref n="10" />,
P Defraigne<fnref n="11" /> ID, E A Donley<fnref n="5" />, P O Hedekvist<fnref n="12" /> ID, I Sesia<fnref n="7" />, M Wouters<fnref n="13" />, P Dubé<fnref n="2" /> ID,
F Fang<fnref n="14" />, F Levi<fnref n="7" />, J Lodewyck<fnref n="4" /> ID, H S Margolis<fnref n="15" /> ID, D Newell<fnref n="5" />, S Slyusarev<fnref n="16" />,
S Weyers<fnref n="6" /> ID, J-P Uzan<fnref n="17" />, M Yasuda<fnref n="18" /> ID, D-H Yu<fnref n="19" />, C Rieck<fnref n="12" />, H Schnatz<fnref n="6" />, Y Hanado<fnref n="8" />,
M Fujieda<fnref n="8" />,<fnref n="21" /> ID, P-E Pottie<fnref n="4" /> ID, J Hanssen<fnref n="20" />, A Malimon<fnref n="16" /> and N Ashby<fnref n="5" /> ID

E-mail: Patrizia.tavella@bipm.org

Received 28 July 2023, revised 3 December 2023

Accepted for publication 21 December 2023

Published 22 January 2024

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CrossMark

Abstract

This paper outlines the roadmap towards the redefinition of the second, which was recently
updated by the CCTF Task Force created by the CCTF in 2020. The main achievements of
optical frequency standards (OFS) call for reflection on the redefinition of the second, but open
new challenges related to the performance of the OFS, their contribution to time scales and
UTC, the possibility of their comparison, and the knowledge of the Earth's gravitational

<figure><img src="image_4.png" /></figure>

<footnote n="1">¹ ARTEMIS, Université Côte d'Azur, Observatoire de la Côte d'Azur, CNRS, Nice, France</footnote>

<footnote n="2">² National Research Council, Ottawa, Canada</footnote>

<footnote n="3">³ Université de Neuchâtel, Neuchâtel, Switzerland</footnote>

<footnote n="4">⁴ LNE-SYRTE, Observatoire de Paris, Université PSL, CNRS, Sorbonne Université, Paris, France</footnote>

<footnote n="5">⁵ National Institute of Standards and Technology, Boulder, CO, United States of America</footnote>

<footnote n="6">⁶ Physikalisch-Technische Bundesanstalt (PTB), Bundesallee, Braunschweig, Germany</footnote>

<footnote n="7">⁷ Istituto Nazionale di Ricerca Metrologica, Turin, Italy</footnote>

<footnote n="8">⁸ National Institute of Information and Communications Technology, Koganei, Tokyo, Japan</footnote>

<footnote n="9">⁹ Bureau International des Poids et Mesures, Sèvres, France</footnote>

<footnote n="10">¹⁰ Emirates Metrology Institute (QCC EMI), Abu Dhabi, United Arab Emirates</footnote>

<footnote n="11">¹¹ Royal Observatory of Belgium, Brussels, Belgium</footnote>

<footnote n="12">¹² Research Institutes of Sweden, Borås, Sweden</footnote>

<footnote n="13">¹³ National Measurement Institute, Sydney, Australia</footnote>

<footnote n="14">¹⁴ National Institute of Metrology, Beijing, People's Republic of China</footnote>

<footnote n="15">¹⁵ National Physical Laboratory, Teddington, United Kingdom</footnote>

<footnote n="16">¹⁶ FSUE VNIIFTRI, Mendeleevo, Solnechnogorsky District, Moscow Region, Russia</footnote>

<footnote n="17">¹⁷ Sorbonne Université, CNRS, UMR 7095, Institut d'Astrophysique de Paris, Paris, France</footnote>

<footnote n="18">¹⁸ National Metrology Institute of Japan, National Institute of Advanced Industrial Science and Technology, Tsukuba, Ibaraki, Japan</footnote>

<footnote n="19">¹⁹ Korea Research Institute of Standards and Science, Daejeon, Republic of Korea</footnote>

<footnote n="20">²⁰ United States Naval Observatory (USNO), Washington, DC, United States of America</footnote>

<footnote n="21">²¹ National Astronomical Observatory of Japan, 2-21-1 Osawa, Mitaka, Tokyo, Japan</footnote>

<footnote>* Author to whom any correspondence should be addressed.</footnote>
potential to ensure a robust and accurate capacity to realize a new definition at the level of $10^{-18}$ uncertainty. The mandatory criteria to be achieved before redefinition have been defined and their current fulfilment level is estimated showing the fields that still needed improvement. The possibility to base the redefinition on a single or on a set of transitions has also been evaluated. The roadmap indicates the steps to be followed in the next years to be ready for a sound and successful redefinition.

Keywords: second, atomic frequency standard, International System of Units

# 1. Introduction

The definitions of the base units of the International System of Units (SI) [1] are decided by the General Conference on Weights and Measures (CGPM) that supervises the work of the International Committee for Weights and Measures (CIPM) and its Consultative Committees. Following definitions based on astronomical phenomena, the definition of the SI unit of time, the second, has relied since 1967 on the caesium atom hyperfine transition frequency (section 2). Caesium primary frequency standards are currently realizing this unit with a relative frequency uncertainty at the low $10^{-16}$ level, but in the last two decades they have been surpassed by optical frequency standards (OFS) showing much lower uncertainties, currently 2 orders of magnitude better.

In 2016, the Consultative Committee for Time and Frequency (CCTF) set up a first version of the roadmap towards the redefinition of the second and the associated conditions for the redefinition [2, 3].

Since June 2020, the roadmap has been updated by a dedicated CCTF Task Force on this topic, with three subgroups related to:

A. Requests from user communities, National Metrology Institutes and Liaisons.

B. Atomic frequency standards, and possible redefinition approaches.

C. Time and Frequency dissemination and time scales.

The CCTF has gathered feedback on the redefinition of the second through a global consultation of concerned communities and stakeholders, which was carried out through an online survey from December 2020 to January 2021. It has analysed the needs and possible impacts of a new definition, not just scientific and technological, but also regulatory and legislative (section 3). The choice of the new definition is central to the debate: the CCTF has analysed the various options that can be envisaged and identified the pros and cons of each possibility (section 4).

The CCTF has updated criteria and conditions that quantify the status of the developments and their maturity for a redefinition (section 5). The fulfilment of mandatory criteria relies on the progress of ultra-low uncertainty and reliable Optical Frequency Standards (OFS—section 6) and Time and Frequency (TF) transfer and comparison techniques (section 7) required for the realization of the new definition and

its dissemination towards users, including the contribution of OFS to the International Atomic Time scale (TAI).

# 2. History of definitions

Until 1967, the SI definition of time had been based on astronomy. It was initially the fraction $1/86 400$ of the mean solar day but observation of unpredictable variations in the Earth rotation rate led in 1960 to a change of the definition to choose a more stable astronomical phenomenon: the motion of the Earth around the Sun, with an SI second equal to the fraction $1/31 556 925.9747$ of the tropical year 1900.

Thanks to the rapid progress of caesium thermal beam frequency standards, the SI definition of the second left the field of astronomy in 1967 to enter the field of quantum physics, with the definition exploiting the benefits of high precision frequency measurements [4]. The second became at that time the ‘the duration of 9192 631 770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the caesium 133 atom’. In 1999, to take black body radiation shifts into account, an addendum to the initial definition was issued to specify that the definition refers to a caesium atom at rest at a temperature of 0 K.

The 26th meeting of the CGPM (2018) marked an important step with the revision of the SI system of units and the redefinition of four base units, by fixing the values of fundamental constants: kilogram (Planck constant $h$), ampere (elementary charge $e$), kelvin (Boltzmann constant $k_B$), and mole (Avogadro constant $N_A$). The basis of the definition of the SI second remained the same but the wording changed in order to be consistent with the general spirit of the new SI, fixing the value of the caesium frequency: ‘The second, symbol s, is the SI unit of time. It is defined by taking the fixed numerical value of the caesium frequency $\Delta\nu_{Cs}$, the unperturbed ground-state hyperfine transition frequency of the caesium-133 atom, to be 9192 631 770 when expressed in the unit Hz, which is equal to $s^{-1}$. In this revised SI, the unit of time has a central position since fixing the values of fundamental constants leads to a direct dependence of all the units, except the mole, on the definition of the second (table 1).

The evolution from astronomy to quantum physics in 1967 was associated with a deep conceptual change for the type of measured quantity underlying the *mise en pratique* of the definition. In astronomy, it was the angle/phase linked to the considered Earth motion that was determined theoretically as a given function of time. With quantum physics, the realization
<table><caption>**Table 1.** Dependencies of the SI base units on defining constants and on other units.</caption><thead><tr><th>Unit</th><th>Defining constant</th><th></th><th>s</th><th>m</th><th>A</th><th>kg</th><th>K</th><th>Cd</th></tr></thead><tbody><tr><td>s</td><td>Δν<sub>Cs</sub>: unperturbed ground-state hyperfine transition<br>frequency of the caesium-133 atom</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>m</td><td>c: speed of light in vacuum</td><td>Dependence on</td><td>X</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>A</td><td>e: elementary charge</td><td>other units</td><td>X</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>kg</td><td>h: Planck constant</td><td></td><td>X</td><td>X</td><td></td><td></td><td></td><td></td></tr><tr><td>K</td><td>k<sub>B</sub>: Boltzmann constant</td><td></td><td>X</td><td>X</td><td></td><td>X</td><td></td><td></td></tr><tr><td>Cd</td><td>K<sub>cd</sub>: luminous efficacy of monochromatic radiation<br>of frequency 540 × 10<sup>12</sup> Hz</td><td></td><td>X</td><td>X</td><td></td><td>X</td><td></td><td></td></tr></tbody></table>

of the definition is now based on frequency measurements, with the assumption provided by the Standard Model that the atomic resonance frequencies are universal and constant, both in time and in space [5–7].

Today, the primary representation of the SI second is realized by caesium primary frequency standards, with relative frequency uncertainties at the 10<sup>-16</sup> level offered by cold atom fountains (see www.bipm.org/en/time-ftp/circular-t and [8]).

Secondary representations of the SI second (SRS) are provided by rubidium or OFS. The list of recommended values of standard frequencies for transitions that may be used as SRS is regularly updated [3, 9, 10].

### 3. Main needs in TF metrology and stimulus for a new definition

With the SI second underlying the realization of other SI units its redefinition may potentially impact a very wide range of communities. Here we consider the impact and the drive for a new definition of the SI second on the metrological community represented by the National Metrology Institutes (NMIs) and the Designated Institutes (DIs), and on the wider timing community. In addition, the findings of the CCTF survey are summarized.

#### 3.1. Significance of the redefinition for the NMIs and DIs

The NMIs and DIs, as part of their mandates, strive to develop the best realizations of the SI units and build the highest accuracy primary standards. They also typically have the most demanding requirements for accessing accurate time and frequency signals because they provide the highest tier SI dissemination services for their respective countries. The current primary frequency standards have now been surpassed in terms of stability and systematic uncertainty by OFS, and, therefore, the NMIs and DIs are expected to drive the transition to the new state-of-the-art definition.

The implementation of a new definition of the SI second, based on optical standards, and an improved Coordinated Universal Time (UTC) will require the metrology labs to acquire new systems and adopt new methods. The stakeholder survey that was conducted in December 2020–January 2021 showed an overall positive response to the redefinition plans, which indicates high levels of commitment and technical maturity that is essential to support the redefinition work.

#### 3.2. Significance of the redefinition for the wider timing community

Although relatively unknown to the general public, sub-µs timing and synchronization capability has become an essential and crucial feature of most critical infrastructure, including telecommunications, energy, finance, cloud computing, transportation and space activities. Even though these applications do not require the accuracies of the optical clocks today, they, in general, depend on TF metrology.

In addition, many scientific applications require nanosecond levels of stability and/or accuracy such as radio astronomy, particle physics experiments, and time metrology. In the next 5–10 y, the need for higher precision in both time and frequency is estimated to grow across all fields.

Initially, the redefinition of the second, and the development in the time and frequency metrology that this may underpin, will mostly support scientific applications, but industrial ones will arise as the related technology becomes available. For example, quantum communications have some time accuracy and stability requirements at the level of femtoseconds, which are hardly achievable with current technologies, and the stimulus of the redefinition may help their development.

#### 3.3. Meeting current and future stakeholder needs

From the CCTF survey and other references [11–14], timing accuracy needs are currently in the range from 1 µs down to 10 ns, while future needs seem to focus below 100 ns for most users. Some scientific users highlighted the need for a sub-nanosecond timing accuracy. The most stringent fractional frequency accuracy needs are currently around 10<sup>-14</sup>, while future needs are specified up to 10<sup>-15</sup> or 10<sup>-18</sup> for some specific users.

The most fundamental of the existing scientific applications that will be improved by a redefinition and the resulting improvement in timing infrastructure, are tests of fundamental physics, for which the levels of accuracy achievable with optical clocks can underpin tests of fundamental physical theories, including the investigation of physics beyond the standard model and time variation of the fundamental constants, the search for dark matter, gravitational wave detection, and more [15].

Better clocks will also enable higher-precision atomic and molecular spectroscopy as well as improved time synchronization for high-resolution telescope arrays and future VLBI
<table><caption>**Table 2.** Stakeholder responses to the question: What level of frequency uncertainty would you like to access in the future?.</caption><thead><tr><td>Uncertainty level</td><td>Application opportunity</td></tr></thead><tbody><tr><td>1 × 10<sup>-14</sup></td><td>Holdover</td></tr><tr><td>1 × 10<sup>-15</sup></td><td>Spectroscopy/dark matter/secure com/holdover</td></tr><tr><td>1 × 10<sup>-16</sup></td><td>Cosmology</td></tr><tr><td>1 × 10<sup>-17</sup></td><td>Dark matter/connected interferometry</td></tr><tr><td>1 × 10<sup>-18</sup></td><td>Positioning/real time geodesy/new clocks</td></tr><tr><td>1 × 10<sup>-19</sup></td><td>Geodynamics</td></tr><tr><td>1 × 10<sup>-20</sup></td><td>Relativistic geodesy/alternative theories of gravitation</td></tr></tbody></table>

generations [16], geopotential monitoring with centimetre resolution [17], quantum networks for quantum encrypted communications [18], and others.

These emerging fields of research that already require better TF accuracy or stability than is available today and applications that promise to transition from the research lab into commercial use in the next decades will benefit from the improved accuracy enabled by a redefinition.

A redefinition of the SI second will also lead to timing infrastructure improvements, including improved time scales and frequency transfer methods. These improvements will benefit the wider stakeholder community, including clock and equipment manufacturers and users. The redefinition of the second constitutes a required step in stabilizing and directing the technology development, standardization and adoption.

Table 2 lists the stakeholder requests for their future needs in the accuracy of frequency references. It is clear from the high level of interest in more accurate frequency reference signals that many research opportunities will arise with better access to optical clocks and better dissemination methods.

## 4. Options for the redefinition of the SI second

The current definition of the SI units is established in terms of a set of seven defining constants with fixed numerical values, as declared in Resolution 1 of the 26th meeting of the CGPM (2018) [19].

Three of these defining constants: *c*, *h*, and *e*, are directly embodied in the fundamental theoretical framework of general relativity and the standard model of particle physics. The defining constant for the unit of time, $\Delta v_{Cs}$, is a property of the Cs atom and consequentially a natural constant. The other three defining constants have a less direct connection to the fundamental framework, $k_B$, $N_A$ being conversion factors, and $K_{cd}$ being linked to the sensitivity of the human eye.

There are three options for the redefinition of the second, which all keep the same principle of applying seven defining constants but would replace $\Delta v_{Cs}$ by a different constant.

Option 1 consists of choosing one single atomic transition in lieu of the Cs hyperfine transition and to fix the numerical value of the frequency of this transition $v_{Xy}$

$$v_{Xy} = N \text{ Hz}, \text{ where } N \text{ is the defining value.}$$

Option 2 consists of creating a defining constant based on several transitions rather than just a single one, as described in [20]. The quantity whose numerical value is used in the definition is a weighted geometrical mean of the frequency of an ensemble of chosen transitions. The unit of time is set by the relation:

$$\prod_i v_i^{w_i} = N \text{ Hz}, \text{ where } w_i \text{ and } N \text{ are the defining values, with the sum of all } w_i \text{ being equal to 1.}$$

Option 3 consists in fixing the numerical value of one more fundamental constant, in addition to *c*, *h* and *e*. From the fundamental standpoint, a good choice for this constant is the electron mass $m_e$ (see e.g. [21]), in which case the system of units is set by the relations:

$$m_e = M \text{ kg},$$

where *M* is the defining value, completed by the other defining relations for *c*, *h*, *e*, $k_B$, $N_A$ and $K_{cd}$.

In this system, one can see that the Compton frequency $v_e$ defined by $h v_e = m_e c^2$ has a defined value, which shows how such a system defines the unit of time. Another choice is to directly fix the numerical value of $v_e$ instead of $m_e$. A third choice is to fix the numerical value of the Rydberg frequency $R_\infty$ which is also linked to the electron mass via the relation $R_\infty = \alpha^2 v_e / 2$, where $\alpha$ is the fine-structure constant. The two first choices are two different formulations for systems of units that are physically identical. The third choice defines a physically different system of units since $\alpha$ is a dimensionless constant that can only be measured and cannot be fixed by our choice.

While all three options concern primarily the definition of the SI second, they would have a formal impact on the definitions of all other base units with the exception of the mole, because these make use of the definition of the second via $\Delta v_{Cs}$.

To complement these formal aspects of the redefinition options, several points are worth noting. Regarding Option 1, it is anticipated that besides the primary transition selected for the definition, other transitions will contribute to realizations and disseminations of the unit of time according to the mechanism of SRS that is already in place and will be described in more detail in section 6. As a possibility associated to Option 2, it is also proposed that future revisions of the defining values $w_i$ and $N$ could be adopted by the CIPM, based on the recommendation of the CCTF and CCU, and according to a set of rules adopted beforehand by the CGPM. Rules include a quantitative criterion to trigger a revision that ensures the convergence through successive updates (see [20]). Rules are designed to ensure that revisions are made only when significant improvement of the realization and dissemination will ensue. This dynamic option is referred to as option 2b, while the option 2 with fixed values of weights and $N$ is named option 2a and the discussion is very much alive on possible unexpected consequences of a 'dynamic' definition;.

Regarding Option 2, the realization makes use of best estimates of optical frequency ratios established via the fitting procedures that are already in place of the CCL-CCTF WGFS
[22]. Given these ratios, one single frequency standard based on either of the chosen transitions can realize the unit of time [20]. In addition to the conceptual aspect, i.e. the possibility to define the unit of time and the system of units using several transitions, Option 2 gives a possible approach to cope with the present context where many different atomic transitions give OFS with uncertainties near $10^{-18}$ and where the field will remain highly dynamic.

Under Option 3, the numerical value of the defining constant for the unit of time relies on experiments that presently lead to the determination of the chosen constant. The evaluations of relevant experiments are the work of CODATA and are reported in [23]. Currently, the value of $m_e$ has an uncertainty of 3.0 parts in $10^{10}$, while the uncertainty in the Rydberg constant is 1.9 part in $10^{12}$. These uncertainties are several orders of magnitude larger than the present realizations of the unit of time of the current SI system (few parts in $10^{16}$) and even further away from the capabilities of OFS ($10^{-18}$ or better). Consequently, Option 3 is not practical in the current state of science and technology.

It is also worth noting that measurements between the optical frequency domain and the current best realizations of the SI second are already done with low enough uncertainty (near $10^{-16}$, the limit of fountain frequency standards) and with sufficient redundancy to ensure the continuity between the current definition and any definition based on optical transitions.

To summarize the trade-offs between the three options, we present here their most significant respective strengths, weaknesses, opportunities, and threats (i.e. a SWOT analysis) in tabular form (table 3). We note that these considerations have taken into account the needs of both the user and research communities, as assessed by the CCTF Task force via input from user surveys and BIPM workshops.

# 5. Criteria and conditions for the redefinition

In order to choose the best new definition and its implementation timeline, and to provide the CGPM with all the required information for making its decision, criteria and conditions (table 4) have been defined to assure that the redefinition:

* Offers an improvement by 10–100 of the uncertainty of the realization of the new definition in the short term after the redefinition (reaching $10^{-17}$–$10^{-18}$ relative frequency uncertainty) and potentially a larger improvement in the longer term (criteria I.1, I.2, III.1 and condition III.3), requiring the capability to compare OFS with an adequate uncertainty to validate OFS uncertainty budgets (criteria II.1, II.2);

* ensures continuity with the current definition based on caesium (criterion I.3);

* ensures continuity and sustainability of the availability of the new SI second through TAI/UTC and enables a significant improvement of the quality of TAI and UTC($k$) as soon

as the definition is changed (criterion I.4 and conditions I.6, III.3), relying on the reliability of OFS and TF transfer infrastructures (conditions I.5, II.3);

* is acceptable to all NMIs and stakeholders and enables the dissemination of the unit to broad categories of users (criterion III.2 and conditions III.4, III.5);

Criteria and conditions are distinguished in the following way:

* the mandatory criteria that must be achieved before changing the definition;

* the ancillary conditions that are not required to be fully achieved to change the definition but are important to ensure the best realization and exploitation of the new definition in the short and long terms. Thus, these conditions correspond to essential work that must have started before the redefinition, with a reasonable amount of progress at the time of redefinition and a commitment of stakeholders to continue their efforts on the associated activities.

Fulfilment indexes have been defined to evaluate the fulfilment level for mandatory criteria to quantitatively follow the improvements, to be aware of the remaining work to fulfil all mandatory criteria and ultimately, to decide it is time to change the definition. The details of criteria and conditions and their current fulfilment levels or progress statuses are presented in section 8.

# 6. OFS—Categories and characteristics

## 6.1. Types, characteristics and performance of OFS

Due to their demonstrated potential for low fractional frequency instabilities and uncertainties, there is currently considerable research activity directed towards investigating optical transitions to serve as frequency standards. These standards fall into two categories distinguished by the charge state of the atom and the method used for trapping: trapped ion optical clocks and optical lattice clocks with neutral atoms. Presently, ten optical transitions and one microwave transition ($^{87}$Rb) are recommended as SRS, as listed in table 5. We note that due to the lower uncertainties associated with most of the optical standards themselves, the uncertainties for the realizations of the second with these standards as listed in the table are largely determined by the uncertainty of microwave standards based on the Cs transition that enters into the recommended frequencies.

Advances in several key technologies have been critical to the rapid improvement in optical standards. To achieve a low instability, it is necessary to start with an extremely narrow linewidth clock laser. Thus, pre-stabilization of the clock laser to a high-performance optical cavity is a standard component of any high-performance standard. Fractional frequency instabilities as low as $8 \times 10^{-17}$ on 1 s timescales have been achieved with a clock laser locked to the resonance frequency
<table><caption>**Table 3.** Collection of Strengths, weaknesses, opportunities and threats of the 3 options for the redefinition, based on input from a community survey in 2022.</caption><thead><tr><th></th><th>Option 1</th><th>Option 2</th><th>Option 3</th></tr></thead><tbody><tr><td>Strengths</td><td>Offers two orders of magnitude improvement of the existing definition with significant improvement likely in the future.<br>Maintains continuity with the current Cs definition.<br>Intuitive extension of the existing definition.<br>Familiar and practical, using primary and secondary realizations as we do today.<br>The unit of time can be realized without additional uncertainty</td><td>Offers two orders of magnitude improvement of the existing definition with significant improvement likely in the future.<br>Maintains continuity with the current Cs definition.<br>Flexible scheme that is well matched to the current experimental situation and could adapt well to rapid progress in optical standards.<br>Could more easily lead to a consensus on the chosen species.</td><td>Consistent with the approach adopted by CIPM based on the physical constants, c, h, e, and k<sub>B</sub><br>Direct connection to the theoretical framework of fundamental physics.</td></tr><tr><td>Weaknesses</td><td>With no clear preferred transition at present, it may be hard to reach a consensus.</td><td>Can be difficult to understand and convey to general users.<br>The unit of time may be hard to realize by a single institute in isolation.<br>The version which allows for revisions of the defining values w<sub>i</sub> and N constitutes a conceptual deviation from the principle of applying fixed defining constants for the SI units as implemented in 2019.<br>A better realization uncertainty of one of the transition alone does not have a full impact on the realization of the unit. Rather all transitions composing the unit should, in average, improve to significantly improve the realization of the unit.<br>The defining constant has no physical meaning—all realizations are secondary representations.<br>A more complex definition of time may present legal issues for some countries.</td><td>Would lead to poor accuracy for time realization in the present and foreseeable future.<br>Would represent a step backwards in time realization by four orders of magnitude (six relative to Options 1 and 2).<br>Would not allow continuity with the current Cs definition, which allows a much better accuracy in the realization.</td></tr><tr><td>Opportunities</td><td>The many benefits associated with an improvement of a factor of 100 (or more) in the definition of the unit of time.<br>A clear path forward for development of primary standards.<br>Provides a stimulus for the development of commercial standards.</td><td>The many benefits associated with an improvement of a factor of 100 (or more) in the definition of the unit of time.<br>Provides a strong stimulus to explore new frequency standard options.</td><td>This approach would lead to a consistent set of SI definitions that is close to the theoretical foundations of physics.<br>Could stimulate further research in simple atoms, calculable quantum systems and the measurements of fundamental constants.</td></tr><tr><td>Threats</td><td>Depending on the quality of future OFS reports for TAI calibration, it might be difficult to provide at least as good uncertainty of dTAI (the fractional frequency deviation of TAI) after the redefinition.<br>The new definition might rapidly become obsolete—SRS could end up dominating contributions to TAI.<br>Could discourage future progress on frequency standards, by biasing work towards the chosen transition.</td><td>Depending on the quality of future OFS reports for TAI calibration, it might be difficult to provide at least as good uncertainty of dTAI after the redefinition.<br>A multi-species definition might lead to difficulty for industry (and NMIs) in choosing which standard to develop.</td><td>There would be a severe degradation in the realization of the SI unit of time.<br>Such a definition would break the metrological principle that redefinitions should be consistent with previous definitions within the uncertainty with which the old definition was realized.</td></tr></tbody></table>
<table><caption>**Table 4.** Mandatory criteria and ancillary conditions to ensure the benefit and the acceptability of a new definition.</caption><thead><tr><th></th><th>Mandatory criteria</th><th>Ancillary conditions</th><th>Criteria and conditions</th></tr></thead><tbody><tr><td rowspan="7">Frequency standards, including the contribution of OFS to time scales</td><td>X</td><td></td><td>I.1—Accuracy budgets of optical frequency standards</td></tr><tr><td>X</td><td></td><td>I.2—Validation of Optical Frequency Standard accuracy budgets—Frequency ratios</td></tr><tr><td>X</td><td></td><td>I.3—Continuity with the definition based on Cs</td></tr><tr><td>X</td><td></td><td>I.4—Regular contributions of optical frequency standards to TAI (as secondary representations of the second)</td></tr><tr><td>X</td><td></td><td>I.5—High reliability of OFS</td></tr><tr><td>X</td><td></td><td>I.6—Regular contributions of optical frequency standards to UTC(k)</td></tr><tr><td></td><td></td><td></td></tr><tr><td rowspan="3">TF links for comparison or dissemination</td><td>X</td><td></td><td>II.1—Availability of sustainable techniques for Optical Frequency Standards comparisons</td></tr><tr><td>X</td><td></td><td>II.2—Knowledge of the local geopotential with an adequate uncertainty level</td></tr><tr><td></td><td>X</td><td>II.3—High reliability of ultra-high stability TF links</td></tr><tr><td rowspan="4">Acceptability of the new definition</td><td>X</td><td></td><td>III.1—Definition allowing more accurate realizations in the future</td></tr><tr><td>X</td><td></td><td>III.2—Access to the realization of the new definition</td></tr><tr><td></td><td>X</td><td>III-3—Continuous improvement of the realization and of time scales after redefinition</td></tr><tr><td></td><td>X</td><td>III.4—Availability of commercial optical frequency standards</td></tr><tr><td></td><td></td><td>X</td><td>III.5—Improved quality of the dissemination towards users</td></tr></tbody></table>

<table><caption>**Table 5.** List of secondary representations of the second adopted by the 22nd CCTF (March 2021) [9].</caption><thead><tr><th>Transition</th><th>Approximate wavelength/nm</th><th>Recommended frequency/Hz</th><th>Recommended relative uncertainty</th><th>Used to calibrate TAI scale interval</th></tr></thead><tbody><tr><td><sup>199</sup>Hg</td><td>265</td><td>1128 575 290 808 154.32</td><td>2.4 × 10<sup>-16</sup></td><td></td></tr><tr><td><sup>27</sup>Al<sup>+</sup></td><td>267</td><td>1121 015 393 207 859.16</td><td>1.9 × 10<sup>-16</sup></td><td></td></tr><tr><td><sup>199</sup>Hg<sup>+</sup></td><td>282</td><td>1064 721 609 899 146.96</td><td>2.2 × 10<sup>-16</sup></td><td></td></tr><tr><td><sup>171</sup>Yb<sup>+</sup>(E2)</td><td>436</td><td>688 358 979 309 308.24</td><td>2.0 × 10<sup>-16</sup></td><td></td></tr><tr><td><sup>171</sup>Yb<sup>+</sup>(E3)</td><td>467</td><td>642 121 496 772 645.12</td><td>1.9 × 10<sup>-16</sup></td><td></td></tr><tr><td><sup>171</sup>Yb</td><td>578</td><td>518 295 836 590 863.63</td><td>1.9 × 10<sup>-16</sup></td><td>Yes (4 institutes)</td></tr><tr><td><sup>88</sup>Sr<sup>+</sup></td><td>674</td><td>444 779 044 095 486.3</td><td>1.3 × 10<sup>-15</sup></td><td></td></tr><tr><td><sup>88</sup>Sr</td><td>698</td><td>429 228 066 418 007.01</td><td>2.0 × 10<sup>-16</sup></td><td></td></tr><tr><td><sup>87</sup>Sr</td><td>698</td><td>429 228 004 229 872.99</td><td>1.9 × 10<sup>-16</sup></td><td>Yes (3 institutes)</td></tr><tr><td><sup>40</sup>Ca<sup>+</sup></td><td>729</td><td>411 042 129 776 400.4</td><td>1.8 × 10<sup>-15</sup></td><td></td></tr><tr><td><sup>87</sup>Rb</td><td></td><td>6834 682 610.904 3126</td><td>3.4 × 10<sup>-16</sup></td><td>Yes (1 institute)</td></tr></tbody></table>

of a room temperature 48 cm ULE FP cavity (ultra-low expansion Fabry-Perot cavity) [24], while locking to cryogenic single-crystal optical cavities has led to frequency instabilities in the low $10^{-17}$ range [25, 26]. In addition, the development of optical frequency combs (OFC) [27, 28], which are needed to link optical frequencies directly with microwave frequencies, has made high-fidelity measurements of absolute optical frequencies at the low $10^{-16}$ uncertainty level of fountain clocks feasible. In fact, simultaneous measurements of the same optical frequency ratio with two independent OFCs have shown agreement at the level of $10^{-21}$ [29], thereby confirming the capability of OFCs to support optical frequency ratio measurements at the limit of the uncertainties of current optical clocks. These capabilities have enabled more precise (and more rapid) comparisons between standards, with many of the optical standards realizing SRS as listed in table 5 reaching Type B uncertainties below $10^{-17}$.

The current record for systematic uncertainty of an atomic clock is held by the $^{27}$Al$^+$ quantum logic clock, with a fractional frequency systematic uncertainty of $9.4 \times 10^{-19}$ [30]. This level of performance is closely followed by that of an Yb optical lattice clock ($1.4 \times 10^{-18}$ [31]), a Sr optical lattice clock ($2.0 \times 10^{-18}$ [32]), an $^{171}$Yb$^+$ ion clock operated on the octupole (E3) transition ($2.7 \times 10^{-18}$ [33, 34]), and recently a $^{40}$Ca$^+$ ion clock ($3.0 \times 10^{-18}$ [35]). Interestingly, it seems there is not a fundamental limitation for the accuracy of the optical clocks that are being developed based on different ion and neutral atom species. Most of the currently proposed optical transitions can potentially achieve an uncertainty level below $10^{-18}$. We note that the lowest instabilities achieved at 1 s averaging time have been observed with optical lattice clocks: $4.8 \times 10^{-17}$ [36] and $6 \times 10^{-17}$ [37]. For single ion clocks, the lowest reported instabilities at 1 s are typically around $1 \times 10^{-15}$ [30, 38].
## 6.2. Ratio measurements between frequency standards

In order to verify the predicted levels of performance for these standards, there has been a great effort over the past decade to perform measurements of frequency ratios between co-located or remotely located standards. Such comparisons can be based on the same transition or different transitions. Comparing different optical standards based on the same transition provides a way to validate uncertainties by verifying that the realized transition frequencies agree within stated uncertainties. To date, several such comparisons performed within the same institute have reached an overall uncertainty better than $5 \times 10^{-18}$ [31, 34], with the lowest reaching $1 \times 10^{-18}$ [31]. Comparisons between standards based on the same transitions from different institutes are at the level of $5 \times 10^{-17}$ [39]. We note that comparisons between clocks in different locations are much more challenging because they involve either remote comparison, which can be limited by the instability of long-distance time transfer capabilities or transportable standards, which generally have lower levels of performance than their lab-based counterparts. In general, such comparisons are of utmost importance to validate the frequency standards' uncertainties.

Equally valuable are frequency ratios measured between standards based on different transitions. Such ratios between unperturbed atomic transitions are significant, because they are dimensionless quantities given by nature. As a result, two independent measurements of such ratios should coincide within the combined measurement uncertainties. Thus, comparisons between independent measurements of given ratios provide further means to validate stated uncertainties of OFS. We note that such measurements almost always rely on optical frequency combs to span the frequency gap between standards. Therefore, comparing independent measurements of a given optical frequency ratio tests not only the stated uncertainties of optical standards themselves, but those of the combs (and any other optical frequency metrology capabilities relevant to the use of OFS). To date, the most accurate measurement of an optical frequency ratio has a fractional uncertainty of $6 \times 10^{-18}$ (between two labs about 2 km apart) [39, 40]. A few optical ratios have been measured multiple times by different institutes, thereby enabling first comparisons of such measurements at uncertainty levels ranging from $3 \times 10^{-17}$ to $2 \times 10^{-16}$.

We also emphasize that frequency ratio measurements between optical and microwave standards are common and serve to validate our capabilities to connect the optical domain with the microwave domain, as well as to link a potential future definition to the current one. The accuracies of such measurements are now at the limit of the primary standards based on Cs (~$10^{-16}$). In the last few years, many such absolute measurements of optical standards have been performed by comparison with TAI, whose rate with respect to the SI second is provided by BIPM publications, based on the currently available reports from primary and secondary frequency standards (www.bipm.org/en/time-ftp/circular-t). Several groups have

performed extended measurement campaigns involving both optical and microwave clocks that have lasted from several months [40–44] to several years [45–47]. Although not continuous, these campaigns were realized by performing multiple measurements over a given time span.

Taken as a whole, the resulting ensemble of high accuracy measurements of atomic frequency ratios published after peer-review provides an overdetermined dataset from which one can determine the best values for these atomic frequency ratios, using an adjustment procedure. This task is done on a regular basis by the CCL-CCTF working group on frequency standards (CCL-CCTF WGFS). The resulting output of this calculation provides the basis for the recommended values and uncertainties of frequency standards shown in table 5 [9]. In addition, given the strongly overdetermined nature of the dataset, this adjustment provides a global validation of the status of high accuracy atomic frequency standards and of related measurement capabilities, as described in [3]. In the last implementation reported to the 22nd meeting of the CCTF on 19 March 2021, the adjustment took into account 105 measurements (69 in 2017), including 33 optical frequency ratios (11 in 2017) and 72 absolute frequency measurements (58 in 2017). We note that it is necessary to take into account correlations (483 for the latest adjustment) between these measurements to perform the calculation correctly [10].

## 6.3. Ongoing research activities and future prospects for optical standards (new transitions, improved stability, transportable standards)

Despite the considerable progress to date in optical clock performance, there remains much room for further improvements in terms of clock stability, uncertainty, and robustness. Reduced clock instability is not only useful in direct timing applications, but the extremely low uncertainty of optical clocks is only useful if the statistical uncertainty (Allan deviation) can be reduced to the evaluated uncertainty level at a practical averaging time for the measurement application. Improvements in the observed stability of optical lattice clocks and long-lived ion transitions ($^{27}$Al$^{+}$, $^{171}$Yb$^{+}$ (E3)) are ongoing but are technically challenging, as they require ultra-stable lasers with coherence times of several seconds to minutes. In addition to continued advances in cavity performance mentioned earlier, there are efforts in parallel to develop novel measurement protocols that mitigate the limitations caused by reference cavity noise, such as zero-dead time interrogation [37], correlation spectroscopy [48, 49], and dynamic decoupling of laser phase noise in compound atomic clocks [50]. It is anticipated that the use of compound clocks could improve the stability of single ion clocks with long clock transition lifetimes to levels comparable to that of optical lattice clocks [50]. For ion species with shorter lifetimes, the stability can be improved directly by increasing the number of ions, but this approach requires special care in the selection of the atomic transition and the control of the systematic shifts to preserve
accuracy [51, 52]. Entanglement in multi-ion or neutral atom clocks offers the potential for a stability beyond the standard quantum limit and thus could be a method to further improve the stability of optical clocks [53]. A new type of clock with high relative stability has been demonstrated recently, called a 'tweezer array optical clock' that balances the benefits of non-interacting particles as found in single-ion clocks with the large number of atoms as found in optical lattice clocks [54].

Another critical aspect for the spread of optical clock performance throughout the clock community will be the demonstration of high duty cycle, high performance, robust optical systems. In this direction there has been considerable effort with many systems under development. Indeed, all major subsystems of an optical clock with laser cooled atoms or ions have already been developed as robust transportable devices for autonomous operation, which have been partially tested for operation in space. This includes vacuum systems and traps for atoms [55] and ions [56], tunable laser systems for cooling and interrogation, optical reference cavities for obtaining a narrow linewidth of the reference laser [57–59], and optical frequency combs for transfer of the optical stability to a microwave output signal [60]. However, the integration of an optical clock from the subsystems also requires the robust optical alignment of multiple laser beams and the monitoring, control and adjustment of a few dozen electrical and mechanical parameters. Fully integrated prototype systems that have been used as transportable optical clocks on the footprint of a small trailer have been demonstrated for a Sr optical lattice clock [61, 62] and for a clock with a single trapped Ca⁺ ion [63].

Some groups have demonstrated high clock operation uptimes, for example 80.3% for a duration of 6 months [42], 93.8% uptime for a period of 10 d [44]. More recently fully autonomous operation for 2 weeks with 99.8% uptime at $2 \times 10^{-17}$ systematic uncertainty inside a laboratory has been demonstrated for the OptiClock based on the E2 transition of $^{171}$Yb⁺ [64, 65]. The system fits inside the volume of two 19 inch racks and has been developed by PTB jointly with industry [65]. Optical clocks with (nearly) 100% uptimes for one month of continuous operation or longer are expected to become common in the next few years. These results indicate that the development of a turn-key autonomous optical clock is technically feasible at a performance level that is superior to available microwave frequency standards and shows the way towards a commercial high-performance optical reference.

Finally, one of the most exciting directions in optical clock research today is the search for transitions that have still lower sensitivities to external fields than current optical clocks in an effort to further reduce clock uncertainties. Some of these include a nuclear transition in $^{229m}$Th [66], and transitions in highly-charged ions [67, 68] and lutetium ions [52]. While all of these systems present their own technological challenges, they could well be among the main candidates for future optical clocks with performance at the 19th and 20th digits.

<figure><img src="image_5.png" /><figcaption>**Figure 1.** Modified Allan deviation of the comparison between IPPP and several other high accuracy techniques: the optical fibre links DTAG-PTB (blue), AOS-GUM (orange) and WR SMD-ESTEC (red) and the two-way carrier phase link NICT-KRIS (green) [Reproduced from [69], with permission from Springer Nature]. WR is the technique White Rabbit Precise Time Protocol [70].</figcaption></figure>

# 7. TF transfer and time scales—categories and characteristics

## 7.1. TF transfer

Remote comparison of time scales and frequency standards has been studied for nearly 50 y using space-based microwave techniques, including Global Navigation Satellite Systems (GNSS) and Two-way Satellite Time and Frequency Transfer (TWSTFT). Besides, another technique using Very Long Baseline Interferometry (VLBI) has been recently realized, although only limited number of laboratories, far from operational network, has demonstrated the capability. In the last decade, optical techniques using fibre optic links have offered greatly improved stability and accuracy. Innovative satellite transfer in the optical domain is also envisaged. Lastly, Transportable OFS or Clocks (TOCs) used as travelling standards can support a redefinition of the second that requires comparisons at an accuracy level of $10^{-18}$ and global geographical coverage.

GNSS time transfer is a one-way technique used since the 1980s, notably for the realization of UTC. A collaboration with the International GNSS Service (IGS) has led to the use of Precise Point Positioning (PPP) for time and frequency comparisons and development of the integer ambiguity PPP technique (IPPP), which to date offers the best long-term stability among the GNSS techniques. Figure 1 [69] shows that IPPP provides time transfer with a modified Allan deviation of $7 \times 10^{-16}/\tau$, where $\tau$ is the averaging time in days of continuous phase measurements. A twofold improvement is
<figure><img src="image_6.png" /><figcaption>**Figure 2.** Modified Allan deviation of UTC(NICT)-UTC(KRIS) from MJD 57851 to 57883 measured by different techniques [Reproduced from [72]. CC BY 3.0.].</figcaption></figure>

expected using satellites from all the GNSS, as opposed to just GPS as at present.

TWSTFT, the second intercontinental-capable satellite-based microwave method, typically employs the code-phase of a signal modulated by a pseudorandom noise code sent and received by microwave link via a geostationary telecommunications satellite, at Ku-band frequencies [71]. Improved performance is achieved by the use of Two-Way Carrier-Phase (TWCP), which exploits carrier-phase measurements, with an instability of a few parts in $10^{16}$ at one day. Further results [72] indicate that TWCP performs at least as well as IPPP in terms of stability. Figure 2 shows the modified Allan deviation of Code Phase and Carrier Phase TWCP.

In addition, a recently implemented software-defined receiver (SDR) successfully reduced the long-term instability by about a quarter [73]. Similar technology is expected to be applied to the transmitters for further improvement resulting in integrated digital modems that are an important step to improve TWSTFT beyond the current state of art. Moreover, in order to reach to the sub 1E-17 level it is essential to improve on modeling of all non-reciprocal error sources, such as signal propagation, atmospheric turbulence, and relativistic effects [74].

VLBI utilizes the reception of radio signals from extragalactic radio sources, with the time difference between the arrivals of the signals measured at two antennas equipped with local atomic clocks. Using VLBI, the frequency of an Yb and a Sr optical standard has been compared [75], with a statistical uncertainty from the VLBI link of $9 \times 10^{-17}$ over 300 h of measurements.

Using optical communication, satellite-based comparisons were demonstrated with the Time Transfer by Laser Link (T2L2), onboard the Jason-2 satellite [76]. Three T2L2 links were compared with IPPP links [77], with the standard deviation of the time difference well below 100 ps. Promising results have also been obtained using terrestrial free-space

optical time and frequency transfer, using CW or coherent pulsed lasers. For both, uncertainties of parts in $10^{16}$ in a few minutes have been achieved over distances up to tens of kilometers. The synchronization of two clocks 28 km apart below 1 fs within 100 s, even at high Doppler velocities of up to $\pm 24 \text{ m s}^{-1}$, and under stable weather conditions has been shown [78]. A comparison at 113 km with modified Allan deviation of $10^{-19}$ at $10^4$ s was also reported [79], the first evidence of the method compatibility with Low Earth Orbit satellites. Figure 3 indicates both results.

Optical fibres offer several key advantages compared to free-space techniques: high isolation from external interference; high bandwidth; and low propagation losses, when compensated by optical amplifiers and regeneration devices, at distances more than 1000 km. For time and frequency comparisons, three main methods are used: CW light from an ultra-stable laser, without modulation; modulated laser light (amplitude, frequency, or phase modulation); and protocol-based signals, based on digital data transfer.

Propagation of optical signals in optical fibres for frequency comparisons offers two main choices: bi-directional fibre links, providing the best performance, and unidirectional fibre links, which are easier to implement on common telecommunication networks. Submarine links are less noisy than terrestrial links [80], as shown in figure 4.

Optical frequency transfer over fully bi-directional links [81] exhibits typical Allan deviations of $\sim 10^{-15}$ at 1 s and $<10^{-18}$ for greater than 100 s, over (100–1000) km long links. There is no systematic frequency shift reported so far at the level of $10^{-18}$. Conversely, optical frequency transfer over unidirectional links has demonstrated an Allan deviation of $\sim 10^{-15}$ at 1 s integration time, unidirectional links have demonstrated an Allan deviation of $7 \times 10^{-17}$ for averaging times between 30 s and 200 s [82]. There is no systematic frequency shift reported so far in the range of $10^{-16}$ [83]. Modulation of the optical carrier frequency enables a frequency reference in the radio and microwave domain (10 MHz–10 GHz) to be transmitted, with typical uncertainty less than $10^{-17}$ at $10^4$ s. Latest synchronization experiments report 300 km free space link and demonstrate a sub ps capability [84].

Time transfer over fibre can be in the radio/microwave domain (10 MHz–10 GHz) or in the optical domain. In either case, the technique requires the modulation (amplitude, phase or frequency) to be tied to a time scale. The time uncertainty is less than 1 ns, approaching tens of picoseconds, in particular with White Rabbit (WR) Precise Time Protocol [70] and the ELSTAB technique [85].

Transportable optical clocks offer the best immediate prospects to meet the criteria for the redefinition of the SI second in regard to the required accuracy level and geographical coverage. As described above, space microwave techniques need to significantly improve their uncertainty levels. Fibre techniques meet the required uncertainty, but obtaining global coverage requires a large effort and investment. Satellite-based optical comparisons have not yet been demonstrated on a full metrological and operational basis. On the other hand,
<figure><img src="image_7.png" /><figcaption>**Figure 3.** Free space optical link fractional frequency instability. Left: Modified Allan deviation over 28 km [Reproduced with permission from [78]. Copyright 2021 American Chemical Society]. *M* is the ratio *f*<sub>r</sub>/Δ*f*<sub>r</sub>, where *f*<sub>r</sub> is the nominal repetition rate, and Δ*f*<sub>r</sub> is the real difference between the repetition rates of the two involved combs. Right: Modified Allan deviation over 113 km [Reproduced from [79], with permission from Springer Nature.] (Black circles, well-aligned free-space time-frequency link; blue squares, mis-aligned link; orange triangles, free-running link). The performances of the best optical clock, the I-SOC (Space Optical Clock on the International Space Station) laser link, the I-SOC microwave link and the TDEV of 1 fs are also shown.</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>**Figure 4.** Submarine testbeds, round-trip phase noise [Reprinted with permission from [80] © The Optical Society]. L indicates land links, S submarine links. Red line: submarine 2 × 96.4 km link; grey line: measurement noise floor; green line: 2 × 150 km fibre along highway; black line: 2 × 92 km fibre along highway (other area).</figcaption></figure>

several TOCs have already reported the performance results that meet the redefinition requirements. An accuracy ranging from $10^{-15}$ down to parts in $10^{18}$ has been reported for several $^{87}$Sr TOCs [86–88]. A bosonic $^{88}$Sr TOC achieved $2 \times 10^{-17}$ uncertainty [89]. TOCs based on ions have also been reported: a $\text{Ca}^{+}$ TOC with a systematic uncertainty of $1.3 \times 10^{-17}$ [90]; an $\text{Al}^{+}$ standard, with four main biases evaluated at the $10^{-18}$ level [91]; and a $\text{Yb}^{+}$ standard demonstrated with $10^{-17}$ accuracy [65]. In addition to their role in the redefinition, the

TOCs are essential tools for chronometric levelling and some have already been used for this purpose [62, 87, 90].

The time and frequency transfer techniques described above allow to compare timescales and their scale intervals around the world. We can also compare the scale intervals by evaluating them with respect to locally available accurate frequency standards. However, this assumes that we have knowledge of the geopotential at the clocks location, since the atomic clocks generate their proper time and the tick rate is affected by the relativistic frequency shift. We should also note that International Atomic Time (TAI) is defined in Resolution 2 of the 26th CGPM (2018) as a realization of Terrestrial Time, which has a reference potential of $W_0$. Thus, the local geopotential needs to be obtained with respect to $W_0$ particularly for the calibration of the TAI scale interval. For the modelling of the geopotential, satellite data only provides information valid at a spatial resolution of 200 km or worse. Combining regional information from gravity measurements with the global model as well as the results of the levelling from the nearest reference to the trapped atoms, the gravity shifts of optical clocks in some metrological laboratories are now evaluated with uncertainty at the mid $10^{-18}$ level or better [92–94].

## 7.2. Time scales

Resolution 2 of the 26th CGPM (2018) states that Coordinated Universal Time (UTC), based on TAI, is the only recommended international time reference and provides the basis of civil time in most countries. Thus, the scale interval of TAI needs to be maintained with respect to OFS for the redefinition of the second. The future TAI should have at least a similar or better performance than the current realization of TAI, which is
nowadays calibrated mainly by microwave-based primary fre-
quency standards. To this end, more than 10 d of regular opera-
tion of optical clocks or a local timescale steered by an optical
clock is required, in each Circular T reporting period, since
the frequency link of local clocks to TAI is made by GNSS or
TWSTFT. For the determination of TAI, the BIPM employs an
uncertainty of ~$10^{-15}/(t/5)$, where *t* is the signal integration
time in days [95].

The capabilities for TAI calibration of several specific OFS
have been examined by the CCTF Working Group on Primary
and Secondary Frequency Standards (CCTF-WGPSFS), and
as a result, eight OFSs, recognized at present as Secondary
Frequency Standards (SFS), have contributed to TAI.

The first data for TAI calibration with an OFS was obtained
in 2014 by an optical ⁸⁷Sr lattice clock from SYRTE [44],
applied for TAI calibration in 2017, and since mid-2021, at
least one SFS has calibrated TAI every month. The BIPM
incorporates the data from SFS into the TAI steering with addi-
tional uncertainty *u*<sub>srep</sub>, which is determined by the uncertainty
in the CIPM recommended frequency of the SFS (table 5). The
recent update of CIPM recommended frequencies has reduced
*u*<sub>srep</sub>, leading to an increased total weight of typically more
than 10% for all SRS for the determination of the TAI scale
interval. The stated uncertainties from the laboratories, ignor-
ing the recommended uncertainty (*u*<sub>srep</sub>) of the SRS, range
from 1.9 × 10<sup>-16</sup>–3.3 × 10<sup>-15</sup>, limited primarily by dead time
and link uncertainties. Until now, the lowest uncertainty in the
SFS data submitted to TAI was reached by the NICT-Sr1 in
*Circular T* 408, and IT-Yb1 in *Circular T* 411.

The calibrations provided from all OFSs [31, 42, 44, 96–99]
are so far consistent with those provided by primary fre-
quency standards (see also https://webtai.bipm.org/database/
d Plot.html). The development of OFSs with high uptimes
over the typical reporting intervals of (15–30) d, the devel-
opment of better local oscillators, and advances in frequency
transfer are crucial goals to obtain significant improvements in
the stability of TAI.

UTC is a post-processed timescale determined by the
BIPM. For civil time, time and frequency metrology laborator-
ies generate and provide real-time signals equivalent to UTC.
These real time signals are called UTC(*k*), denoting a real-time
UTC generated at the laboratory ‘*k*’. In general, such a UTC(*k*)
is often employed as a national standard time with the addition
of a time offset appropriate to the respective time zone. For the
future redefinition of the second, UTC(*k*) generated or at least
steered by an optical clock is one ancillary condition. UTC(*k*)
time scales must be continuous, whereas it is unrealistic at this
point to operate optical clocks completely without dead time.
The operation of multiple optical clocks for redundancy is not
yet realized since maintaining multiple optical clocks is a dif-
ficult task and the procedure to switch between optical clocks
has not yet been studied. On the other hand, intermittent opera-
tion of an optical clock enables generation of a real-time times-
cale steered by the optical clock [100–104]. Here, the source
oscillator is still a microwave oscillator (hydrogen maser), but
the scale interval is tuned with respect to an optical clock.
In some metrology institutes, a similar generation of UTC(*k*)

has already been successfully implemented for some time
utilizing caesium fountain frequency standards [105–108]. In
future, an all-optical timescale is expected [109], particularly
for improvement of the short-term stability. Here, a CW laser,
stabilized to a stable optical cavity, would play the role of
the source oscillator. Considerable progress in mode-hop free
operation of CW lasers has been made in the last decade.

**8. Fulfilment levels of mandatory criteria—Progress status for ancillary conditions**

Details on mandatory criteria and ancillary conditions are
presented in sections 8.1–8.3 for OFS, TF transfer and the
acceptability of a new definition. A synthesis of the fulfil-
ment levels of mandatory criteria in 2022 is shown in figure 5.
Fulfilment regions have been defined, from very low fulfilment
levels (<30%, region in red) to satisfactory fulfilment levels
(90%–100% and above, region in green). The vertical dashed
blue line defines the threshold above which the criteria can be
considered as fulfilled.

While for certain criteria the fulfilment seems almost
achieved, for others the fulfilment is more challenging. Due
to the considerable number of OFS under development, good
progress has been made on OFS performance Criteria I.1 and
I.3 (fulfilment levels close to 50% and 100%, respectively) and
on their contributions to TAI Criterion I.4 (fulfilment level of
30%–50%). Regardless of which redefinition option is chosen,
the realizations of the definition will be accessible widely and
their accuracy will likely continue to improve in the future
with further developments on OFS (Criteria III.1 and III.2).
However, the challenges associated with limited resources for
developing multiple standards in one institute (along with lim-
itations in long distance time transfer) have led to a low fulfil-
ment level (<30%) for the OFS’ comparison Criterion I.2.

The fulfilment of the criterion II.2 related to the knowledge
of the geopotential is achieved in the majority of NMIs oper-
ating an OFS.

For the criteria II.1, a sustainable technique for OFS com-
parison at the proper uncertainty level is more challenging.
Over intracontinental scales (baselines of about 1000 km), the
requirement is fulfilled by optical fibre links, even if a sig-
nificant effort for regular comparison campaigns should be
addressed.

8.1. *Criteria and conditions related to OFS and their contribution to time scales*

This section contains a detailed description of the criteria and
the estimation of their fulfilment level in 2022.

**Mandatory criterion I.1—Accuracy budgets of OFS**

I.1.a—At least three OFS based on the same reference
transition, in different institutes, have demonstrated eval-
uated relative frequency uncertainties ≲2 × 10⁻¹⁸ based
on comprehensive, comparable and published accuracy
budgets.

Fulfilment level: 20%–40% [30–32].
<figure><img src="image_9.png" /><figcaption>**Figure 5.** Fulfilment levels of mandatory criteria in 2022.</figcaption></figure>

I.1.b—At least three frequency evaluations of OFS based on different reference transitions, either in the same institute or different institutes, have demonstrated evaluated uncertainties ≲2 × 10⁻¹⁸ based on comprehensive, comparable and published accuracy budgets.

Fulfilment level: 80%–100% [30–32].

→ Overall Fulfilment level of criterion I.1: 30%–50%.

**Mandatory criterion I.2—Validation of optical frequency standard accuracy budgets—Frequency ratios**

I.2.a—Unit ratios (frequency comparison between standards with same clock transition): at least three measurements between OFS in different institutes in agreement with an overall uncertainty of the comparison Δν/ν ≲5 × 10⁻¹⁸ (either by transportable clocks or advanced links). Applicable to at least one radiation of I.1.

Fulfilment level: 0%–20% [31, 34, 62].

Strictly speaking the reported measurements of unit ratios are not between different institutes and should not count in this fulfilment level. Nevertheless, a fulfilment level at 0%–20% has been assigned based on these in house comparisons with uncertainties significantly lower than 5 × 10⁻¹⁸ that can be considered as the first step in the right direction.

I.2.b—Non unit ratios (frequency comparison between standards with different clock transitions): at least five measurements between standards among I.1 or other, each ratio measured at least twice by different institutes in agreement with an overall uncertainty of the comparison Δν/ν <5 × 10⁻¹⁸ (either by direct comparisons, transportable clocks or advanced links).

Fulfilment level: 0%–20% [40].

Again, this measurement alone is not valid in terms of the criterion which demands ratio measurements

«twice by independent» institutes. However, it is the first measurement at about the required uncertainty level, and it is considered the first step towards the fulfilment of this index.

→ Overall Fulfilment level for Criterion I.2: <30%.

**Mandatory criterion I.3—Continuity with the definition based on Cs**

There are at least three independent frequency evaluations of the optical frequency transitions utilized by the standards in I.1) with TAI or with three independent Cs primary frequency standards (in different or the same institutes), possibly via optical frequency ratio measurements, where the measurements are limited essentially by TAI or by the uncertainty of these Cs frequency standards (Δν/ν <3 × 10⁻¹⁶).

→ Fulfilment level: 90%–100% [44–47, 97, 98, 104, 110, 111].

**Mandatory criterion I.4—Regular contributions of OFS to TAI (as secondary representations of the second)**

At least three state-of-art calibrations of TAI (uncertainty ≲2 × 10⁻¹⁶ without counting the recommended uncertainty of the secondary representation of the second *u*<sub>srep</sub>) each month from a set of at least five OFS for at least 1 y. Check that there is no degradation of TAI if its calibrations were done by OFS considered as primary standards and Cs frequency standards considered as secondary standards.

Fulfilment level: 30%–50% [see www.bipm.org/en/time-ftp/circular-t, and https://webtai.bipm.org/database/show_psfsf.htm, https://webtai.bipm.org/database/d Plot.html].

**Ancillary condition I.5—High reliability of OFS**

Reliable continuous operation capability of OFS, in a laboratory environment, with the appropriate level of uncertainty.
Progress status: Typical uptimes of OFS over measurement durations >10 d currently cover a wide range from a few percent to 90% [44, 111, 112], and www.bipm.org/en/time-ftp/circular-t.

### Ancillary condition I.6—Regular contributions of OFS to UTC(k)

Progress status: Preliminary tests of UTC(k) steered by an OFS [100–103, 109].

## 8.2. Criteria and conditions related to TF links for comparison or dissemination

### Mandatory criterion II.1—Availability of sustainable techniques for optical frequency standard comparisons

Availability and sustainability of transportable clocks or TF links with uncertainties <5 × 10⁻¹⁸ for frequency comparisons between at least NMIs operating OFS of I.1), on a national/intracontinental basis (baseline up to about 1000 km). Capability of repeated uncertainty estimations of these links.
→ Fulfilment level: 50%–70% [62, 113, 114].

### Mandatory criterion II.2—Knowledge of the local geopotential with an adequate uncertainty level

Knowledge of geopotential differences for NMIs operating OFS of I.2) to be consistent with the uncertainty budget of a frequency comparison between OFS using advanced links, i.e. including the uncertainty budget of the two OFS and of the link. Knowledge of local geopotential for NMIs operating OFS of I.4) with an uncertainty corresponding to a frequency uncertainty ≲10⁻¹⁷, for the calibration of TAI.
→ Fulfilment level: 70%–90% [39, 87, 92–94] and www.bipm.org/en/time-ftp/data.

### Ancillary condition II.3—High reliability of ultra high stability TF links

On-demand continuous operation capability of TF links over sufficient durations that do not limit OFS comparisons and their regular contributions to TAI.

Progress Status: a few months continuous operation of fibre links for intracontinental comparisons [113, 115] but no existing link allowing OFS intercontinental comparisons without degradation.

## 8.3. Criteria and conditions related to the acceptability of the new definition

### Mandatory criterion III.1—Definition allowing future more accurate realizations

The new definition must be long lasting. On the short term (just after the redefinition), it must ensure an improvement by 10/100 of its realization with OFS, i.e. reaching 10⁻¹⁷/10⁻¹⁸ relative frequency uncertainty. On the longer term, it must have the potential for further improvement of the realization of 10⁻¹⁸ and beyond in order to avoid any early obsolescence of the definition.

→ Fulfilment level: 100% (To be confirmed, based on the chosen option for the redefinition, but no identified fundamental effect limiting OFS accuracy at 10⁻¹⁸ level for

all species in I.1, and some newer systems have the potential to go beyond 10⁻¹⁸)

### Mandatory criterion III.2—Access to the realization of the new definition

III.2.a Realization/‘mise en pratique’ of the new definition must be easily understandable with a clear uncertainty evaluation process;

Fulfilment level: 0% (No existing document; pending the choice of the redefinition option)

III.2.b—Access for NMIs and high accuracy users to primary or secondary realizations of the new definition;
Fulfilment level: 100% (To be confirmed, based on the chosen option for the redefinition, but primary or secondary representations of the SI second will continue to be accessible via metrology institutes or TAI)

III.2.c—Cs frequency standards ensure a secondary realization of the new definition.

Fulfilment level: 100% (existing TAI architecture will be maintained at current level or better and Cs will be a secondary representation of the second)

→ Overall Fulfilment level for Criterion III.2: 70%–90%.

### Ancillary condition III.3—Continuous improvement of the realization and of time scales after redefinition

Commitment of NMIs to make the best effort to:

* improve and operate OFS that provide primary or secondary realizations of the new definition (reliable/continuous operation, regular contributions to TAI, ...);

Progress status: Several OFS are already in operation and used by the CCL-CCTF Working Group on Frequency Standards (CCL-CCTF-WGFS) to calculate the Recommended values of standard frequencies 2021 [10]

* maintain the operation of Cs fountain standards over the appropriate duration;

Progress status: 12 Cs fountains in operation [116–124]

* development of new OFS;

Progress status: Several other atomic species are being investigated as potential candidates for the next generation, for example ²²⁹Th⁺, Lu⁺, Cd, and several highly charged ions. The most recent references can be found for example in [Proceedings of the annual IEEE IFCS https://ieee-uffc.org/symposia/ifcs, and EFTF conferences www.eftf.org/].

### Ancillary condition III.4—Availability of commercial OFS

Progress status: No available commercial OFS.

### Ancillary condition III.5—Improved quality of the dissemination towards users

Progress status of TF links (GNSS, TWSTFT, Fibre/Internet) for the dissemination of the definition towards users:

* Frequency stability: 10⁻¹⁷–10⁻¹⁶ for satellite microwave techniques (GNSS, TWSTFT); 10⁻²⁰ level for fibre links [125]

* Time accuracy: 1 ns for satellite microwave techniques (GNSS, TWSTFT); 50 ps for fibre links [126].
<figure><img src="image_10.png" /><figcaption>**Figure 6.** Scenarios for the roadmap initially discussed by the CCTF in 2021. The scenario in the centre is now the baseline.</figcaption></figure>

## 9. Schedule, conclusions, and perspectives

The possible redefinition scenarios depend on capabilities of OFS and their envisaged evolution, considering their performance, their readiness for sustainable contributions to the realization of time scales, especially TAI, and also their potential for commercial availability, and space qualification. A roadmap also needs to address TF transfer techniques required for the comparison of atomic clocks, for the construction of international time scales, and for the dissemination of reference signals to users, with an adequate uncertainty level.

Depending on the achievements and the development progress, the CCTF initially discussed the possible three schedule options for the redefinition (figure 6).

It appeared clear that a redefinition at the 28th meeting of the CGPM (2026) was unrealistic since today there is no consensus on the preferred option and still some important work to do to fulfil all mandatory criteria. The 28th CGPM (2026) could validate a roadmap towards a redefinition in 2030 if, in 2026, there is a consensus on the redefinition option to be chosen and if the work to fulfil mandatory criteria is likely to be achievable by 2030. If a redefinition is not possible in 2030, it will have to be postponed until the meeting of the CGPM to be held in 2034 or the following one. But, with this third scenario, it will require the continued operation of Cs fountains primary frequency standards until the late 2030s.

The redefinition will be the occasion to further educate stakeholders on the concept of metrological traceability and

the best practices for accuracy and stability measurements and their specification. The CCTF will set up a subgroup to address this particular matter and educate the public about the redefinition.

In November 2022, the 27th CGPM approved Resolution 5 [127] corresponding to the CCTF roadmap towards the redefinition of the second as presented in this paper, with a preferred scenario leading to a redefinition at the 29th CGPM (2030). This scenario is realistic, even if there is still considerable work to converge on a preferred option and to fulfil all mandatory criteria by pushing the limits of OFS and T/F transfer. All these efforts will be determining factors in reaching the goal of a new definition of the SI second with an improved quality of the *mise en pratique*, in order to serve current and future needs in metrology and to foster scientific and technological applications at the highest accuracy.

### Data availability statement

No new data were created or analysed in this study.

### Authors contribution

This paper is based on the work of the CCTF Task Force on the 'Roadmap to the redefinition of the second'. The Task Force was chaired by N Dimarcq, P Tavella, and formed by three subgroups, whose members are listed below.
<table><thead><tr><th>Subgroup</th><th>Chairs</th><th>Executive secretary</th><th>Members</th></tr></thead><tbody><tr><td>A Request from user communities, NMIs, and Liaisons</td><td>M Gertsvolf, G Mileti</td><td>F Meynadier</td><td>J Bartholomew, P Defraigne, E A Donley, P O Hedekvist, I Sesia, M Wouters</td></tr><tr><td>B Atomic frequency standards, and possible redefinition approaches</td><td>S Bize, C W Oates, E Peik</td><td>G Petit</td><td>P Dubé, F Fang, T Ido, F Levi, J Lodewyck, H S Margolis, D Newell, S Slyusarev, S Weyers, J-P Uzan, M Yasuda, D-H Yu</td></tr><tr><td>C TF Dissemination T Ido and time scales</td><td>D Calonico</td><td>G Panfilo</td><td>P Defraigne, E A Donley, M Fujieda, M Gertsvolf, Y Hanado, J Hanssen, H S Margolis, G Petit, P-E Pottie, C Rieck, H Schnatz, A Malimon, M Wouters, N Ashby</td></tr></tbody></table>

## ORCID iDs

M Gertsvolf https://orcid.org/0000-0002-1188-2104

S Bize https://orcid.org/0000-0003-2483-5152

T Ido https://orcid.org/0000-0003-3500-6042

P Tavella https://orcid.org/0000-0002-7505-7314

F Meynadier https://orcid.org/0000-0003-2719-5592

G Petit https://orcid.org/0000-0002-6837-3052

P Defraigne https://orcid.org/0000-0001-7780-8540

P O Hedekvist https://orcid.org/0000-0003-0801-3124

P Dubé https://orcid.org/0000-0003-0133-2165

J Lodewyck https://orcid.org/0000-0003-0808-6964

H S Margolis https://orcid.org/0000-0002-8991-3855

S Weyers https://orcid.org/0000-0003-4484-6481

M Yasuda https://orcid.org/0000-0001-6040-3176

M Fujieda https://orcid.org/0000-0002-9934-7586

P-E Pottie https://orcid.org/0000-0003-3677-2208

N Ashby https://orcid.org/0000-0003-0719-3979

## References

[1] BIPM The International System of Units (SI) 9th edn (available at: www.bipm.org/documents/20126/41483022/SI-Brochure-9-EN.pdf)

[2] CCTF 2016 Strategy Document (available at: www.bipm.org/documents/20126/35554894/CCTF+Strategy/7cf0f648-2afe-d15c-0909-1f03406bbb8f)

[3] Riehle F, Gill P, Arias F and Robertsson L 2018 The CIPM list of recommended frequency standard values: guidelines and procedures Metrologia 55 188

[4] Leschiutta S 2005 The definition of the ‘atomic’ second Metrologia 42 S10

[5] Uzan J P 2011 Varying constants, gravitation and cosmology Living Rev. Relativ. 14 1–155

[6] Martins C J A P 2017 The status of varying constants: a review of the physics, searches and implications Rep. Prog. Phys. 80 126902

[7] Will C M 2014 The confrontation between general relativity and experiment Living Rev. Relativ. 17 4

[8] Petit G, Arias F and Panfilo G 2015 International atomic time: status and future challenges C. R. Physique 16 480–8

[9] CCL CCTF Recommended values of standard frequencies (available at: www.bipm.org/en/publications/mises-en-pratique/standard-frequencies)

[10] Margolis H S, Panfilo G, Petit G, Oates C, Ido T and Bize S The CIPM list of recommended frequency standard values: 2021 update (in progress)

[11] EUSPA 2021 Report on Time & Synchronisation User Needs and Requirements, Annex 5 (available at: www.gse.europa.eu/sites/defaultTFiles/sites/all/files/Report_on_User_Needs_and_Requirements_Timing_Synchronisation.pdf)

[12] GIANO consortium 2020 GIANO project (available at: www.euspa.europa.eu/simplecount.pdf/tracker?file=uploads/ucp2020_giano_galileo-based_timing_receiver_for_critical_infrastructures.pdf)

[13] GSA GNSS Market Report 2019 (available at: www.euspa.europa.eu/2019-gsa-gnss-market-report)

[14] Cybersecurity and Infrastructure Security Agency 2020 Report on positioning, navigation, and timing (PNT) backup and complementary capabilities to the global positioning system (GPS) (available at: www.cisa.gov/sites/default/files/publications/report-on-pnt-backup-complementary-capabilities-to-gps_508.pdf)

[15] Safronova M S 2019 Atomic clocks: the search for variation of fundamental constants with clocks Ann. Phys. 531 1800364

[16] Clivati C et al 2020 Common-clock very long baseline interferometry using a coherent optical fiber link Optica 7 1031–1037

[17] Mehlstaubler T E, Grosche G, Lisdat C, Schmidt P O and Denker H 2018 Atomic clocks for geodesy Rep. Prog. Phys. 81 064401

[18] Duan L-M, Lukin M D, Cirac J I and Zoller P 2001 Long-distance quantum communication with atomic ensembles and linear optics Nature 414 413–8

[19] Resolution 1 of the 26th CGPM 2018 On the revision of the International System of Units (SI) (available at: www.bipm.org/en/committees/cg/cgpm/26-2018/resolution-1)

[20] Lodewyck J 2019 On a definition of the SI second with a set of optical clock transitions Metrologia 56 055009

[21] Bordé C 2019 A consistent unified framework for the new system of units: matter-wave optics C. R. Physique 20 22–32

[22] Joint working group of the Consultative Committee for Length (CCL) and the Consultative Committee for Time and Frequency (CCTF) CCL-CCTF working group on frequency standards Terms of reference as well as information on the fitting procedure to establish the list of recommended values of frequency standards and of secondary representation of the second (available at: www.bipm.org/en/committees/cc/cctf/wg/ccl-cctf-wgfs)

[23] Tiesinga E, Mohr P J, Newell D B and Taylor B N 2021 CODATA recommended values of the fundamental physical constants: 2018 Rev. Mod. Phys. 93 025010

[24] Häfner S, Falke S, Grebing C, Vogt S, Legero T, Merimaa M, Lisdat C and Sterr U 2015 8 × 10⁻¹⁷ fractional laser frequency instability with a long room-temperature cavity Opt. Lett. 40 2112
[25] Robinson J M, Oelker E, Milner W R, Zhang W, Legero T, Matei D G, Riehle F, Sterr U and Ye J 2019 Crystalline optical cavity at 4 K with thermal-noise-limited instability and ultralow drift *Metrologia* **6** 240–3

[26] Matei D G et al 2017 1.5 µm lasers with sub-10 mHz linewidth *Phys. Rev. Lett.* **118** 263202

[27] Jones D J, Diddams S A, Ranka J K, Stentz A, Windeler R S, Hall J L and Cundiff S T 2000 Carrier-envelope phase control of femtosecond mode-locked lasers and direct optical frequency synthesis *Science* **288** 635–40

[28] Diddams S A, Jones D J, Ye J, Cundiff S T, Hall J L, Ranka J K, Windeler R S, Holzwarth R, Udem T and Hänsch T W 2000 Direct link between microwave and optical frequencies with a 300 THz femtosecond laser comb *Phys. Rev. Lett.* **84** 5102–5

[29] Johnson L A M, Gill P and Margolis H S 2015 Evaluating the performance of the NPL femtosecond frequency combs: agreement at the $10^{-21}$ level *Metrologia* **52** 62–71

[30] Brewer S M, Chen J-S, Hankin A M, Clements E R, Chou C W, Wineland D, Hume D and Leibrandt D 2019 $^{27}$Al$^+$ quantum-logic clock with a systematic uncertainty below $10^{-18}$ *Phys. Rev. Lett.* **123** 033201

[31] McGrew W F et al 2018 Atomic clock performance enabling geodesy below the centimetre level *Nature* **564** 87–90

[32] Bothwell T, Kedar D, Oelker E, Robinson J M, Bromley S L, Tew W L, Ye J and Kennedy C J 2019 JILA SrI optical lattice clock with uncertainty of $2.0 \times 10^{-18}$ *Metrologia* **56** 065004

[33] Huntemann N et al 2016 Single-ion atomic clock with $3 \times 10^{-18}$ systematic uncertainty *Phys. Rev. Lett.* **116** 063001

[34] Sanner C, Huntemann N, Lange R, Tamm C, Peik E, Safronova M S and Porsev S G 2019 Optical clock comparison for Lorentz symmetry testing *Nature* **567** 204

[35] Huang Y, Zhang B, Zeng M, Hao Y, Ma Z, Zhang H, Guan H, Chen Z, Wang M and Gao K 2022 Liquid-nitrogen-cooled Ca$^+$ optical clock with systematic uncertainty of $3 \times 10^{-18}$ *Phys. Rev. Appl.* **17** 034041

[36] Oelker E et al 2019 Demonstration of $4.8 \times 10^{-17}$ stability at 1 s for two independent optical clocks *Nat. Photon.* **13** 714–9

[37] Schioppo M et al 2017 Ultrastable optical clock with two cold-atom ensembles *Nat. Photon.* **11** 48–52

[38] Dörscher S, Huntemann N, Schwarz R, Lange R, Benkler E, Lipphardt B, Sterr U, Peik E and Lisdat C 2021 Optical frequency ratio of a $^{171}$Yb$^+$ single-ion clock and a $^{87}$Sr lattice clock *Metrologia* **58** 015005

[39] Lisdat C et al 2016 A clock network for geodesy and fundamental science *Nat. Commun.* **7** 12443

[40] Beloy K et al 2019 Boulder Atomic Clock Optical Network (BACON) Collaboration 2021 Frequency ratio measurements at 18-digit accuracy using an optical clock network *Nature* **591** 564

[41] Takano T, Takamoto M, Ushijima I, Ohmae N, Akatsuka T, Yamaguchi A, Kuroishi Y, Munekane H, Miyahara B and Katori H 2016 Geopotential measurements with synchronously linked optical lattice clocks *Nat. Photon.* **10** 662–6

[42] Kobayashi T, Akamatsu D, Hosaka K, Hisai Y, Wada M, Inaba H, Suzuyama T, Hong F-L and Yasuda M 2020 Demonstration of the nearly continuous operation of an $^{171}$Yb optical lattice clock for half a year *Metrologia* **57** 065021

[43] Leopardi H et al 2021 Measurement of the $^{27}$Al$^+$ and $^{87}$Sr absolute optical frequencies *Metrologia* **58** 015017

[44] Lodewyck J et al 2016 Optical to microwave clock frequency ratios with a nearly continuous strontium optical lattice clock *Metrologia* **53** 1123

[45] Schwarz R et al 2020 Long term measurement of the $^{87}$Sr clock frequency at the limit of primary Cs clocks *Phys. Rev. Res.* **2** 033242

[46] Lange R et al 2021 Improved limits for violations of local position invariance from atomic clock comparisons *Phys. Rev. Lett.* **126** 011102

[47] Nemitz N et al 2021 Absolute frequency of $^{87}$Sr at $1.8 \times 10^{-16}$ uncertainty by reference to remote primary frequency standards *Metrologia* **58** 025006

[48] Takamoto M, Takano T and Katori H 2011 Frequency comparison of optical lattice clocks beyond the Dick limit *Nat. Photon.* **5** 288–92

[49] Clements E R, Kim M E, Cui K, Hankin A M, Brewer S M, Valencia J, Chen J-S, Chou C-W, Leibrandt D R and Hume D B 2020 Lifetime-limited interrogation of two independent $^{27}$Al$^+$ clocks using correlation spectroscopy *Phys. Rev. Lett.* **125** 243602

[50] Dörscher S, Al-Masoudi A, Bober M, Schwarz R, Hobson R, Sterr U and Lisdat C 2020 Dynamical decoupling of laser phase noise in compound atomic clocks *Commun. Phys.* **3** 185

[51] Keller J, Burgermeister T, Kalincev D, Didier A, Kulosa A P, Nordmann T, Kiethe J and Mehlstäubler T E 2019 Controlling systematic frequency uncertainties at the $10^{-19}$ level in linear Coulomb crystals *Phys. Rev. A* **99** 013405

[52] Zhiqiang Z, Arnold K J, Kaewuam R and Barrett M D 2023 $^{176}$Lu$^+$ clock comparison at the $10^{-18}$ level via correlation spectroscopy *Sci. Adv.* **9** eadg1971

[53] Pedrozo-Peñafiel E et al 2020 Entanglement on an optical atomic-clock transition *Nature* **588** 414–8

[54] Young A W, Eckner W J, Milner W R, Kedar D, Norcia M A, Oelker E, Schine N, Ye J and Kaufman A M 2020 Half-minute-scale atomic coherence and high relative stability in a tweezer clock *Nature* **588** 408–413,2020

[55] Elliott E R et al 2018 NASA's Cold Atom Lab (CAL): system development and ground test status *NPJ Microgravity* **4** 16

[56] Burt E, Prestage J D, Tjoelker R L, Enzer D G, Kuang D, Murphy D W, Robison D E, Seubert J M, Wang R T and Ely T A 2021 Demonstration of a trapped-ion atomic clock in space *Nature* **595** 43–47

[57] Leibrandt D R et al 2011 Field-test of a robust, portable, frequency-stable laser *Opt. Express* **19** 10278

[58] Vogt S, Lisdat C, Legero T, Sterr U, Ernsting I, Nevsky A and Schiller S 2011 Demonstration of a transportable 1 Hz-linewidth laser *Appl. Phys. B* **104** 741–5

[59] Argence B, Prevost E, Lévèque T, Le Goff R, Bize S, Lemonde P and Santarelli G 2012 Prototype of an ultra-stable optical cavity for space applications *Opt. Express* **20** 25409

[60] Lezius M et al 2016 Space-borne frequency comb metrology *Optica* **3** 1381–1387

[61] Koller S B et al 2017 Transportable optical lattice clock with $7 \times 10^{-17}$ uncertainty *Phys. Rev. Lett.* **118** 073601

[62] Takamoto M, Ushijima I, Ohmae N, Yahagi T, Kokado K, Shinkai H and Katori H 2020 Test of general relativity by a pair of transportable optical lattice clocks *Nat. Photon.* **14** 411

[63] Cao J, Zhang P, Shang J, Cui K, Yuan J, Chao S, Wang S, Shu H and Huang X 2017 A compact, transportable single-ion optical clock with $7.8 \times 10^{-17}$ systematic uncertainty *Appl. Phys. B* **123** 112

[64] Ritter S et al 2020 Opticlock: transportable and easy-to-operate optical single-ion clock *OSA Quantum 2.0 Conf. Optical Society of America* p QTh5B.6
[65] Stuhler J et al 2021 Opticlock: transportable and easy-to-operate optical single-ion clock *Meas. Sens.* **18** 100264

[66] Thielking J, Okhapkin M V, Głowacki P, Meier D M, von der Wense L, Seiferle B, Düllmann C E, Thirolf P G and Peik E 2018 Laser spectroscopic characterization of the nuclear-clock isomer $^{229}$mTh *Nature* **556** 321–5

[67] Kozlov M G, Safronova M S, López-Urrutia J C and Schmidt P O 2018 Highly charged ions: optical clocks and applications in fundamental physics *Rev. Mod. Phys.* **90** 045005

[68] Micke P, Leopold T, King S A, Benkler E, Spieß L J, Schmöger L, Schwarz M, Crespo López-Urrutia J R and Schmidt P O 2020 Coherent laser spectroscopy of highly charged ions using quantum logic *Nature* **578** 60–65

[69] Petit G 2021 Sub-$10^{-16}$ accuracy GNSS frequency transfer with IPPP GPS Solut. **25** 22

[70] Serrano J et al 2009 The White Rabbit Project *Proc. of ICALEPCS TUC004* (Kobe, Japan) (available at: https://accelconf.web.cern.ch/icalepcs2009/papers/tuc004.pdf)

[71] Piestier D, Bauch A, Breakiron L, Matsakis D, Blanzano B and Koudelka O 2008 Time transfer with nanosecond accuracy for the realization of International Atomic Time *Metrologia* **45** 185–98

[72] Fujieda M et al 2018 Advanced satellite-based frequency transfer at the $10^{-16}$ level *IEEE Trans. Ultrason. Ferroelectr. Freq. Control* **65** 973–8

[73] Jiang Z et al 2018 Use of software-defined radio receivers in two-way satellite time and frequency transfers for UTC computation *Metrologia* **55** 685

[74] Geršl J 2023 Relativistic theory for time and frequency transfer through flowing media with an application to the atmosphere of Earth *Astron. Astrophys.* **673** A144

[75] Pizzocaro M et al 2021 Intercontinental comparison of optical atomic clocks through very long baseline interferometry *Nat. Phys.* **17** 223–7

[76] Samain E, Exertier P, Courde C, Fridelance P, Guillemot P, Laas-Bourez M and Torre J-M 2015 Time transfer by laser link: a complete analysis of the uncertainty budget *Metrologia* **52** 423

[77] Leute J et al 2018 High accuracy continuous time transfer with GPS IPPP and T2L2 *Proc. 2018 Europ. Freq. and Time Forum* pp 249–52 (available at: https://ieeexplore.ieee.org/document/8409043)

[78] Ellis J L, Bodine M I, Swann W C, Stevenson S A, Caldwell E D, Sinclair L C, Newbury N R and Deschênes J-D 2021 Scaling up frequency-comb-based optical time transfer to long terrestrial distances *Phys. Rev. Appl.* **15** 034002

[79] Shen Q et al 2022 Free-space dissemination of time and frequency with $10^{-19}$ instability over 113 km *Nature* **610** 661–6

[80] Clivati C, Tampellini A, Mura A, Levi F, Marra G, Galea P, Xuereb A and Calonico D 2018 Optical frequency transfer over submarine fiber links *Optica* **5** 893–901

[81] Predehl K et al 2012 A 920-kilometer optical fiber link for frequency metrology at the 19th decimalplace *Science* **336** 441–4

[82] Schioppo M et al 2022 Comparing ultrastable lasers at $7 \times 10^{-17}$ fractional frequency instability through a 2220 km optical fibre network *Nat. Commun.* **13** 212

[83] Xu D, Lopez O, Amy-Klein A and Pottie P-E 2020 Unidirectional two-way optical frequency comparison and its fundamental limitations *Opt. Lett.* **45** 6074–7

[84] Caldwell E D, Deschenes J-D, Ellis J, Swann W C, Stuhl B K, Bergeron H, Newbury N R and Sinclair L C

2023 Quantum-limited optical time transfer for future geosynchronous links *Nature* **618** 721–6

[85] Krehlik P, Sliwczynski L, Buczek L, Kolodziej J and Lipinski M 2016 ELSTAB-fiber-optic time and frequency distribution technology: a general characterization and fundamental limits *IEEE Trans. Ultrason. Ferroelectr. Freq. Control* **63** 993–1004

[86] Ohmae N et al 2021 Transportable strontium optical lattice clocks operated outside laboratory at the level of $10^{-18}$ uncertainty *Adv. Quantum Technol.* **4** 2100015

[87] Grotti J et al 2018 Geodesy and metrology with a transportable optical clock *Nat. Phys.* **14** 437–41

[88] Kong D-H, Wang Z-H, Guo F, Zhang Q, Lu X-T, Wang Y-B and Chang H 2020 A transportable optical lattice clock at the National Time Service Center *Chin. Phys.* **B** **29** 070602

[89] Origlia S et al 2018 Towards an optical clock for space: compact, high-performance optical lattice clock based on bosonic atoms *Phys. Rev.* A **98** 053443

[90] Huang Y et al 2020 Geopotential measurement with a robust, transportable $Ca^+$ optical clock *Phys. Rev. A* **102** 050802

[91] Hannig S, Pelzer L, Scharnhorst N, Kramer J, Stepanova M, Xu Z T, Spethmann N, Leroux I D, Mehlstäubler T E and Schmidt P O 2019 Towards a transportable aluminum ion quantum logic optical clock *Rev. Sci. Instrum.* **90** 053204

[92] Riedel F et al 2020 Direct comparisons of European primary and secondary frequency standards via satellite techniques *Metrologia* **57** 045005

[93] Denker H, Timmen L, Voigt C, Weyers S, Peik E, Margolis H S, Delva P, Wolf P and Petit G 2018 Geodetic methods to determine the relativistic redshift at the level of $10^{-18}$ in the context of international timescales: a review and practical results *J. Geod.* **92** 487–516

[94] Pavlis N K and Weiss M A 2017 A re-evaluation of the relativistic redshift on frequency standards at NIST, Boulder, Colorado, USA *Metrologia* **54** 535

[95] Panfilo G and Parker T E 2010 A theoretical and experimental analysis of frequency transfer uncertainty, including frequency transfer into TAI *Metrologia* **47** 552

[96] Hachisu H, Petit G, Nakagawa F, Hanado Y and Ido T 2017 SI-traceable measurement of an optical frequency at low $10^{-16}$ level without a local primary standard *Opt. Express* **25** 8511

[97] Kim H, Heo M-S, Park C Y, Yu D-H and Lee W-K 2021 Absolute frequency measurement of the $^{171}$Yb optical lattice clock at KRISS using TAI for over a year *Metrologia* **58** 055007

[98] Pizzocaro M, Bregolin F, Barbieri P, Rauf B, Levi F and Calonico D 2020 Absolute frequency measurement of the $^{1}S_0$-$^{3}P_0$ transition of $^{171}$Yb with a link to International Atomic Time *Metrologia* **57** 035007

[99] Hobson R et al 2020 A strontium optical lattice clock with $1 \times 10^{-17}$ uncertainty and measurement of its absolute frequency *Metrologia* **57** 065026

[100] Hachisu H, Nakagawa F, Hanado Y and Ido T 2018 Months-long real-time generation of a time scale based on an optical clock *Sci. Rep.* **8** 4243

[101] Formichella V, Galleani L, Signorile G and Sesia I 2021 Robustness tests for an optical time scale *Metrologia* **59** 015002

[102] Zhu L, Lin Y, Wang Y, Jia Z, Wang Q, Li Y, Yang T and Fang Z 2022 Preliminary study of generating a local time scale with NIM $^{87}$Sr optical lattice clock *Metrologia* **59** 055007

[103] Yao J et al 2019 Optical-clock-based time scale *Phys. Rev. Appl.* **12** 044069
[104] Grebing C, Al-Masoudi A, Dörscher S, Häfner S, Gerginov V, Weyers S, Lipphardt B, Riehle F, Sterr U and Lisdat C 2016 Realization of a timescale with an accurate optical lattice clock *Optica* **3** 563–569

[105] Bauch A, Weyers S, Piester D, Staliuniene E and Yang W 2012 Generation of UTC(PTB) as a fountain-clock based time scale *Metrologia* **49** 180

[106] Rovera G D, Bize S, Chupin B, Guena J, Laurent P, Rosenbusch P, Uhrich P and Abgrall M 2016 UTC(OP) based on LNE-SYRTE atomic fountain primary frequency standards *Metrologia* **53** S81

[107] Galleani L, Signorile G, Formichella V and Sesia I 2020 Generating a real-time time scale making full use of the available frequency standards *Metrologia* **57** 065015

[108] Whibberley P, English E L, Langham C, Szymaniec K and Hendricks R 2019 Improvement of the UTC(NPL) time scale recent upgrades and future plans *Joint Conf. IEEE Int. Frequency Control Symp. and European Frequency and Time Forum (EFTF/IFC)* (Orlando, FL, USA) pp 1–2

[109] Milner W R *et al* 2019 Demonstration of a timescale based on a stable optical carrier *Phys. Rev. Lett.* **123** 173201

[110] McGrew W F *et al* 2019 Towards the optical second: verifying optical clocks at the SI limit *Optica* **6** 448–54

[111] Cao J *et al* 2022 A compact, transportable optical clock with $1 \times 10^{-17}$ uncertainty and its absolute frequency measurement *Appl. Phys. Lett.* **120** 054003

[112] Hill I R *et al* 2015 8th symposium on frequency standards and metrology 2015 *J. Phys.: Conf. Ser.* **723** 012019

[113] Cantin E *et al* 2021 An accurate and robust metrological network for coherent optical frequency dissemination *New J. Phys.* **23** 053027

[114] Akatsuka T *et al* 2020 Optical frequency distribution using laser repeater stations with planar lightwave circuits *Opt. Express* **28** 9186–97

[115] Guillou-Camargo F *et al* 2018 First industrial-grade coherent fiber link for optical frequency standard dissemination *Appl. Opt.* **57** 7203–10

[116] Li R, Gibble K and Szymaniec K 2011 Improved accuracy of the NPL-CsF2 primary frequency standard: evaluation of distributed cavity phase and microwave lensing frequency shifts *Metrologia* **48** 283

[117] Guéna J *et al* 2012 Progress in atomic fountains at LNE-SYRTE *IEEE Trans. Ultrason. Ferroelectr. Freq. Control* **59** 391

[118] Domnin Y S, Baryshev V N, Boyko A I, Elkin G A, Novoselov A V, Kopylov L N and Kupalov D S 2013 The MTsR-F2 fountain-type cesium frequency standard *Meas. Tech.* **55** 1155

[119] Levi F, Calonico D, Calosso C E, Godone A, Micalizio S and Costanzo G A 2014 Accuracy evaluation of ITCsF2: a nitrogen cooled cesium fountain *Metrologia* **51** 270

[120] Fang F, Li M, Lin P, Chen W, Liu N, Lin Y, Wang P, Liu K, Suo R and Li T 2014 NIM5 Cs fountain clock and its evaluations *Metrologia* **52** 454

[121] Weyers S, Gerginov V, Kazda M, Rahm J, Lipphardt B, Dobrev G and Gibble K 2018 Advances in the accuracy, stability, and reliability of the PTB primary fountain clocks *Metrologia* **55** 789

[122] Jallageas A, Devenoges L, Petersen M, Morel J, Bernier L G, Schenker D, Thomann P and Südmeyer T 2018 First uncertainty evaluation of the FoCS-2 primary frequency standard *Metrologia* **55** 366

[123] Beattie S, Jian B, Alcock J, Gertsvolf M, Hendricks R, Szymaniec K and Gibble K 2020 First accuracy evaluation of the NRC-FCs2 primary frequency standard *Metrologia* **57** 035010

[124] Takamizawa A, Yanagimachi S and Hagimoto K 2022 First uncertainty evaluation of the cesium fountain primary frequency standard NMIJ-F2 *Metrologia* **59** 035004

[125] Raupach S M, Koczwara A and Grosche G 2015 Brillouin amplification supports $1 \times 10^{-20}$ uncertainty in optical frequency transfer over 1400 km of underground fiber *Phys. Rev. A* **92** 021801

[126] Śliwczyński Ł, Krehlik P, Kołodziej J, Schnatz H, Piester D, Bauch A, Imlau H and Ender H 2019 Calibrated optical time transfer of UTC(k) for supervision of telecom networks *Metrologia* **56** 015006

[127] General Conference on Weights and Measures (CGPM) 2022 Resolution 5 of the 27th CGPM *On the future redefinition of the second* (available at: www.bipm.org/en/cgpm-2022/resolution-5)