# On a definition of the SI second with a set of optical clock transitions

Jérôme Lodewyck  LNE–SYRTE, Observatoire de Paris, Université PSL, CNRS,
Sorbonne Université, 61 avenue de l’Observatoire, F-75014 Paris, France
<EMAIL_ADDRESS>
###### Abstract

The current SI second based on the atomic hyperfine transition in the ground
state of 133Cs is expected to be replaced by a new definition based on optical
frequency standards, whose estimated uncertainty has now been established two
orders of magnitude lower than the accuracy of the best Cs primary standards.
However, such a redefinition of the second is hindered by the fact that many
atomic species are potential contenders to become the new primary frequency
standard. In this paper, we propose to resolve this issue by defining a
composite frequency unit based on the weighted geometric mean of the
individual frequencies of different atomic transitions. This unit has the
property to be realisable with any single clock whose transition composes the
unit, provided that at least a few frequency ratios are available, with an
accuracy that marginally differs from the nominal clock uncertainty. We show
that the unit can be updated as the performances of the contributing
transitions evolve, without incurring a drift on the unit itself.

##  1 Introduction

Since 1967, the definition of the Système International (SI) unit of time, the
second, has been based on the hyperfine energy splitting of the ground state
of 133Cs atoms [1]. The initial definition, adopted at the 13th Conférence
Générale des Poids et Mesures (CGPM) states that:

> The second is the duration of 9 192 631 770 periods of the radiation
> corresponding to the transition between the two hyperfine levels of the
> ground state of the cesium 133 atom.

Since then, the accuracy of best realisations of this definition, that is to
say the uncertainty with which the unit is realised, has improved by one order
of magnitude every decade, from the first cesium beam clocks to the modern
cold atom fountain clocks. To take into account the evolution of these cesium
standards, now reaching a control of systematic effects down to a few part in
$10^{-16}$, the definition of the SI second was amended in 1997:

> This definition refers to a caesium atom at rest at a temperature of 0 K.

The 26th CGPM in 2018 introduced a new wording of the definition of the time
unit, in order to make it consistent with the newly adopted definitions of the
other base units:

> The second, symbol s, is the SI unit of time. It is defined by taking the
> fixed numerical value of the caesium frequency $\Delta\nu_{\textrm{\tiny
> Cs}}$, the unperturbed ground-state hyperfine transition frequency of the
> caesium 133 atom, to be 9 192 631 770 when expressed in the unit Hz, which
> is equal to s-1.

The SI is thus formally defined by fixing fundamental constants, among which
$\Delta\nu_{\textrm{\tiny Cs}}$:

> The International System of Units, the SI, is the system of units in which
> the unperturbed ground state hyperfine transition frequency of the caesium
> 133 atom $\Delta\nu_{\textrm{\tiny Cs}}$ is 9 192 631 770 Hz […]

In this major update of the SI, the definition of the second is therefore left
unchanged. However, since the late 2000s, clocks based on optical transitions
in ions or neutral atoms have been developed with a control of systematic
effects better than the accuracy of the best Cs standards [2], raising the
question of a redefinition of the SI second [3, 4, 5, 6]. A roadmap crafted by
the Comité Consultatif Temps Fréquence (CCTF) of the Bureau International des
Poids et Mesures (BIPM) and summarized in [7], fixes milestones that the
optical clocks must reach in order to make such a redefinition possible.
Beyond the realisation of optical clocks with a low uncertainty, the roadmap
emphasizes on the necessity to connect the new unit of time to the current SI
second, to show the reproducibility of the optical clocks across different
laboratories, and to measure frequency ratios between different optical
transitions with an uncertainty matching the uncertainty of the best clocks.
The progresses in completing these milestones let us envision a possible
redefinition of the SI second as early as 2026. However, while the choice of
cesium in 1967 was univocal111Retrospectively, the choice of cesium turned out
to be the correct one as, in 2019, Cs is still one the most appropriate
species for a low uncertainty microwave clock, equally matched with rubidium.,
current optical clocks are being developed with various atomic species, and
various technologies. This diversity of high performance optical clocks is a
wealth for the community, because it fosters applications of precise metrology
in the field of fundamental physics. But it also means that none of these
optical transitions currently stands out as an obvious choice for a new
definition of the SI second, and the fast evolution of the field of optical
frequency metrology makes picking a single atomic species uncertain. Facing
this situation, it seems unavoidable that a new definition of the SI second
should not designate a specific atomic transition as the new standard, but
rather define the frequency unit from a weighted mix of the best realised
optical transitions. This article proposes such a definition for a unit of
time that can accommodate with the multiplicity of frequency standards and
their evolution with time.

##  2 Current status of the Cs atom in the SI

Figure 1: Measured frequency ratios, adapted from [7]. Green labels: secondary
representations of the second (SRS). Blue: other optical transitions; Black
lines: absolute frequency measurements against Cs; Red lines: optical-to-
optical frequency ratios; Orange lines: optical-to-microwave frequency ratios
against the Rb SRS.

Cesium is still the unique primary frequency standard (PFS) with which the SI
second can be realised. However, over the past decade, the frequency ratios
between cesium clocks and various other microwave or optical clocks have been
precisely measured, with uncertainties limited by the best cesium fountains in
the low $10^{-16}$. From these published measurements, it becomes practically
feasible to realise the SI second with a very low uncertainty with other
atomic species. This possibility is enacted by the Comité International des
Poids et Mesures (CIPM), via the publication of recommended frequencies for
so-called secondary representations of the SI second (SRS). The current system
of primary and secondary representations of the SI seconds is reproduced in
table 1.

Frequency/Hz | Fractional uncertainty | Transition
---|---|---
9 192 631 770 | exact |  133Cs ground state hfs
6 834 682 610.904 3126 | $6\times 10^{-16}$ |  87Rb ground state hfs
429 228 004 229 873.0 | $4\times 10^{-16}$ |  87Sr 5s2 1S0 – 5s5p 3P0
444 779 044 095 486.5 | $1.5\times 10^{-15}$ |  88Sr+ 5s 2S1/2 – 4d 2D5/2
518 295 836 590 863.6 | $5\times 10^{-16}$ |  171Yb 6s2 1S0 – 6s6p 3P0
642 121 496 772 645.0 | $6\times 10^{-16}$ |  171Yb+ 2S1/2 – 2F7/2
688 358 979 309 308.3 | $6\times 10^{-16}$ |  171Yb+ 6s 2S1/2 – 5d 2D3/2
1064 721 609 899 145.3 | $1.9\times 10^{-15}$ |  199Hg+ 5d106s 2S1/2 – 5d96s2 2D5/2
1121 015 393 207 857.3 | $1.9\times 10^{-15}$ |  27Al+ 3s2 1S0 – 3s3p 3P0
1128 575 290 808 154.4 | $5\times 10^{-16}$ |  199Hg 6s2 1S0 – 6s6p 3P0
Table 1: Table of the frequencies of the Cs primary and secondary
representation of the SI second, adopted at the 2017 CIPM under the advise of
the CCTF. Reproduced from [7]

More recently, several frequency ratios that do not involve the cesium clock
transition have also been measured, some of them with a relative uncertainty
lower than the accuracy of the best cesium clocks (see figure 1). To
incorporate these measurements in the recommended values of the SRS, an
adjustment procedure using a least square algorithm or graph theory [8, 9] is
now implemented by the CCTF. While apparently preserving the central role of
cesium as the primary standard – the recommended frequencies of the SRS are
given in the SI unit Hz – these procedures actually make a step towards a Cs-
free, decentralized, definition of the frequency unit. For instance, computing
the ratio between the recommended frequencies, in Hz, of the strontium and
mercury optical clock transitions gives an estimate of the real value of this
ratio with an uncertainty lower than the accuracy of the cesium fountain
clocks. From this point of view, the outcome of the adjustment procedures is
not a set of frequencies in Hz, but more fundamentally a set of best estimates
for the ratios $\rho_{ij}$ between the frequencies $\nu_{i}$ of the connected
atomic species:

$\rho_{ij}=\frac{\nu_{i}}{\nu_{j}}\qquad\textrm{with}\quad\rho_{ij}=\rho_{ik}\rho_{kj}.$
(1)

The transitivity condition stated at the end of equation (1) is true, by
definition, for the actual frequency ratios. However, a set of physical
measurements of these ratios may fail to verify the transitivity relation
because of systematic and statistical uncertainty in the measurements. To
circumvent this issue, and to ensure that the best estimates for frequency
ratios are transitive, the adjustment procedures of [8, 9] set the
transitivity relations as fixed constraints. In this picture, the numbers
$\nu_{i}^{\textrm{\tiny SRS}}$ listed in the first column of table 1 can be
formally reformulated as the transitive rational frequency ratio matrix in the
(133Cs, 87Rb, 87Sr, 88Sr+, 171Yb, 171Yb+, 171Yb+, 199Hg+, 27Al+, 199Hg) basis:

$\rho^{\textrm{\tiny SRS}}_{ij}=\frac{\nu_{i}^{\textrm{\tiny
SRS}}}{\nu_{j}^{\textrm{\tiny SRS}}},\quad\textrm{and the }^{133}\textrm{Cs
ground state hfs is }9\,192\,631\,770~{}\textrm{Hz}.$ (2)

This matrix representation explicitly highlights that the recommendations for
the SRS are in essence independent of the Cs frequency, the latter being
factorized by the instantiation of the definition of the SI second. The
uncertainty matrix on the recommended frequency ratios would then read:

$\delta\rho^{\textrm{\tiny SRS}}_{ij}=\rho^{\textrm{\tiny
SRS}}_{ij}\sqrt{u_{i}^{2}+u_{j}^{2}},$ (3)

where $u_{i}$ is the relative uncertainty of the realisation of the clock
transition $i$. In this equation, we assumed that the uncertainty on the
frequency ratios is limited by the systematic uncertainty of the clocks, and
therefore is not improved by accumulating frequency ratio measurements or by
exploiting the redundancy between the coefficients of the frequency ratio
matrix222In practice, the uncertainty given by equation (3) may not be met.
Exploiting statistical independence in the measurement of frequency ratios can
lead to a lower uncertainty. On the other hand, if the frequency ratio
$\rho_{ij}$ was not directly measured but deduced from the product
$\rho_{ik}\rho_{kj}$, the uncertainty may be larger than the uncertainty given
by equation (3). Such cases will be considered later in this paper.. While
equation (2) is equivalent to the list of frequencies of table 1, the
uncertainty matrix of equation (3), although it is not currently published
along with the adjustment of SRS frequencies chosen by the CCTF, contains more
information than the second column of table 1, because it also incorporates
the uncertainty of frequency ratios not including the Cs clock transitions.

Because it does not directly focus on a single atomic transition, the matrix
representation of the recommendations published by the CIPM, given by
equations (2) and (3), can be used to construct a new definition of the SI
second that can be realised not only with a clock based on a specific,
arbitrarily chosen transition, but rather by any high performance optical
frequency standard. For this, we just have to reformulate the anchor to the Cs
stated in the second part of equation (2). This is the aim for the next
sections.

##  3 Notations

We note $\nu_{i}$ the physical frequency of the clock transition $i$. This is
a quantity with a physical dimension, and as such, it is not experimentally
accessible without a prior definition of the frequency unit. On the other
hand, the frequency ratios $\rho_{ij}$ between these frequencies, as defined
by equation (1) can be experimentally measured by comparing clocks. We note
$\rho^{\textrm{\tiny m}}_{ij}$ the outcomes of such measurements. They
generally do not form a complete frequency ratio matrix as pairs of clock
transitions may have never been directly compared, nor do they satisfy the
transitivity relation, as they typically deviate from the genuine frequency
ratios by the uncertainty of the clocks. However, we shall assume that they
are connected, _i.e._ there are no subsets of the frequency ratio measurements
that do not share at least one clock transition. This requirement is necessary
in order to be able to compute, from the measurements, a most likely frequency
ratio matrix that is both complete and transitive [8, 9]. We note
$\bar{\rho}_{ij}$ such a frequency ratio matrix, of which the matrix
$\rho^{\textrm{\tiny SRS}}_{ij}$ is an example.

As stated above, $u_{i}$ is the uncertainty of the clocks based on the
transition $i$. To ease the reading, we sometimes call $u_{i}$ the uncertainty
of transition $i$.

##  4 Requirements for a unit

We consider a frequency unit constructed from the frequencies of various
atomic clock transitions. Such a frequency unit $\nu$ is mathematically
defined by a function $F$ of $n$ individual frequencies $\nu_{i}$:

$\nu=F(\nu_{1},\ldots,\nu_{n}).$ (4)

For instance, the current SI frequency unit is defined by

$\nu^{\textrm{\tiny SI}}\equiv 1~{}\textrm{Hz}=\frac{\nu_{\textrm{\tiny
Cs}}}{N^{\textrm{\tiny SI}}}\quad\textrm{with}\ N^{\textrm{\tiny
SI}}=9\,192\,631\,770.$ (5)

Such a definition must fulfill two _sine qua non_ requirements. The frequency
unit must be:

  * •

realisable in the form of a physical signal, _i.e._ there must exist a
practical device, accessible to various laboratories, whose output is a
realisation of the frequency unit with an inaccuracy as low as possible.

  * •

backward compatible with the previous definitions of the unit, _i.e._ the
relative frequency difference between a new definition of the unit and the
previous definition must be lower than the accuracy of the best realisation at
the moment of the change of definition.

The current definition of the SI second based on Cs fulfills these
requirements: It is realisable with a cold atom atomic fountain clock, and the
constant $N^{\textrm{\tiny SI}}$ was chosen so that the atomic second defined
with Cs would match the previous definition of the SI second based on the
ephemeris time [10].

Beyond these requirements, additional properties can be desirable for a
frequency unit. The frequency unit should be:

  * •

optimal: no other unit can be better realised than the chosen frequency unit.

  * •

universal: a clock based on any transition $i$ should be sufficient to realise
the frequency unit within the uncertainty of the clock.

  * •

representative: the lower the uncertainty of a transition, the larger the
weight of the frequency of this transition in the definition of the unit.

  * •

evolutive: the definition evolves with the uncertainties of the various clock
transitions.

With the advent of various optical clocks with uncertainties smaller than the
accuracy of the cesium fountain clocks, the current definition of the SI
second does not fulfill anymore these properties.

##  5 Construction of a weighted frequency unit

We now propose to construct a new frequency unit suitable for optical
transitions. For this, we write a general form for the definition of the unit
of frequency as the arithmetic mean of the frequencies of different atomic
clock transitions belonging to the set $\mathcal{C}$:

$\nu=\sum_{i\in\mathcal{C}}a_{i}\nu_{i}.$ (6)

The question then arises of choosing the best values for the coefficients
$a_{i}$, both to ensure that the unit can be realised with the lowest possible
uncertainty, and to allow for these coefficients to evolve with time as the
uncertainty of the various transitions composing the unit improves, without
introducing a drift in the unit. Let us first express that transitions should
contribute with different weights $w_{i}$, based on their uncertainty. A
transition with a lower uncertainty will have a larger weight than a
transition with a higher uncertainty. Taking into account that each transition
has its own nominal frequency, this yields the relation between the $a_{i}$
coefficients:

$a_{i}\rho_{ij}/a_{j}=w_{i}/w_{j}$ (7)

for all pairs $(i,j)$, which is a relation symmetric with the transformation
$i\leftrightarrow j$ given that $\rho_{ij}=1/\rho_{ji}$. It is straightforward
to check that the $a_{i}$ coefficients defined as:

$a_{i}=\frac{1}{N}w_{i}\prod_{k\in\mathcal{C}}\rho_{ki}^{w_{k}},$ (8)

where $N$ is a numerical constant, satisfy equation (7), provided the weights
$w_{i}$ are normalised:

$\sum_{k\in\mathcal{C}}w_{k}=1.$ (9)

Then, the frequency unit can be rewritten as:

$\nu=\sum_{i\in\mathcal{C}}a_{i}\nu_{i}=\sum_{i\in\mathcal{C}}\frac{1}{N}w_{i}\prod_{k\in\mathcal{C}}\rho_{ki}^{w_{k}}\nu_{i}=\frac{1}{N}\sum_{i\in\mathcal{C}}w_{i}\prod_{k\in\mathcal{C}}\nu_{k}^{w_{k}}=\frac{1}{N}\prod_{k\in\mathcal{C}}\nu_{k}^{w_{k}}$
(10)

The frequency unit is thus the weighted geometric mean of the frequencies of
the clock transitions. This property fits with the fact that, in essence, the
ratios between frequencies are quantities more representative than the
frequencies themselves, which are conventional and unit-dependent. This fact
is also used in the fitting procedure of reference [9], through the use of the
logarithms of frequency ratios.

##  6 Definition of the frequency unit

Using the results of the previous section, we can now formally define a unit
of frequency as the weighted geometric mean of the frequencies of a set
$\mathcal{C}$ of clock transitions:

$\nu=\frac{1}{N}\prod_{i\in\mathcal{C}}\nu_{i}^{w_{i}},$ (11)

where $\nu_{i}$ are the individual transition frequencies, $N$ is a numerical
constant, and $w_{i}$ the normalized weight of the clock transition $i$. The
normalization of the weights ensures that $\nu$ has the dimension of a
frequency. The current Cs based SI unit of frequency, as expressed by equation
(5), is a special case of equation (11) with $N=N^{\textrm{\tiny SI}}$ and
$w_{i}=\delta_{i,\textrm{\tiny Cs}}$.

The frequency unit (11) is realisable with a single clock based on any
transition $i$, and thus universal, through the use of the frequency ratio
matrix:

$\left.\nu\right|_{i}=\frac{\nu_{i}}{N_{i}}\qquad\textrm{with}\quad
N_{i}=N\prod_{k\in\mathcal{C}}\bar{\rho}_{ik}^{w_{k}},$ (12)

where $\bar{\rho}$ is a measurement-based estimate of $\rho$ satisfying the
transitivity property, for instance the matrix $\rho^{\textrm{\tiny SRS}}$
constructed from the recommended frequency of the SRS already published by the
CIPM. $N_{i}$ is the factor linking the frequency of the clock transition $i$
to the frequency unit: it can be viewed as the recommended frequency for the
clock transition $i$. The $N_{k}$ numbers satisfy the relation:

$\frac{N_{i}}{N_{j}}=\bar{\rho}_{ij}.$ (13)

Then, the relative uncertainty on the realisation of the frequency unit with a
clock based on transition $i$ is given by

$\left.\frac{\delta\nu}{\nu}\right|_{i}=\sqrt{\left(\frac{\delta
N_{i}}{N_{i}}\right)^{2}+u_{i}^{2}}.$ (14)

This uncertainty is composed of the uncertainty $u_{i}$ of the clock used to
realise the unit, but also of the uncertainty on the independently determined
quantity $N_{i}$. The frequency unit is thus non-optimal as soon as the
definition incorporates a transition beside the best transition of the moment.

##  7 Trade-off optimality vs. universality

The quantity $N_{i}$ required to realise the unit with a clock based on
transition $i$ can be computed from a compilation of world-wide frequency
ratio measurements. Its uncertainty $\delta N_{i}$ thus depends on which
ratios are available, on the uncertainty of the clocks that were used to
measure them, but also on the correlations between these ratios, that can
occur if systematic effects shift the frequency of different clocks in the
same way, or if some ratios were measured with the same clocks, possibly at
the same time. A best estimate of $N_{i}$ and its uncertainty $\delta N_{i}$
can be extracted from a least-squares fit that takes as input the set of
measured frequency ratios and their correlations, as explained in [8]. The A
of this paper explicitly details this calculation.

In this section, we consider the relative uncertainty $\delta N_{i}/N_{i}$ for
a few ideal cases, in order to theoretically quantify the deviation from
optimality of the unit proposed in the previous section. We show that an
appropriate choice of weighs $w_{i}$ leads to a realisation of the unit with a
limited sub-optimality, by both accurate and less accurate clocks, resulting
in a universal unit without significantly compromising its optimality.

###  7.1 $n$ clock transitions with identical uncertainties

We first assume the unit is composed of $n$ clock transitions, whose
frequencies are realised by clocks with the same relative uncertainty $u$, and
for which all frequency ratios have been evaluated with a relative uncertainty
$\sqrt{2}u$. The weights of the clock transitions in the unit should therefore
be identical, _i.e._ $w_{i}=1/n$. In this case, the deviation of the unit from
optimality, characterized by $\delta N_{i}/N_{i}$, is at most the clock
uncertainty $u$ if the ratio measurements are fully correlated, or goes to
zero like $1/\sqrt{n}$ for $n$ large if the frequency ratio measurements are
uncorrelated (see A). In practice, a set of measured frequency ratios would
make $\delta N_{i}/N_{i}$ lay between these two bounds, such that the relative
uncertainty on the realisation of the unit for a clock based on any transition
$i$ would satisfy:

$u<\left.\frac{\delta\nu}{\nu}\right|_{i}<\sqrt{2}u.$ (15)

It differs at most by a factor $\sqrt{2}$ from optimality, and approaches
optimality when a large number of uncorrelated frequency ratio measurements
are available.

###  7.2 Partial frequency ratio matrix

Figure 2:  A transition $n+1$ is added to a pool of $n$ initial transitions.
The frequency ratios between this new transition and the initial transitions
is only available for a single initial transition, the “connected transition”.
This figure shows the uncertainty of the realisation of the frequency unit for
the connected transition (thin lines) and unconnected transitions (thick
lines), as a function of the weight $w_{n+1}$ of the newly introduced
transition, from 0 (the unit does not incorporate the new transition) to
$w_{1,n}$ (the new transition has the same weight as the initial transitions).
The measured frequency ratios are assumed to be uncorrelated.

We now assume that another transition with the same uncertainty $u$ is added
to the pool of transitions, bringing the total number of transitions to $n+1$.
However, the connection of this new transition is not complete, _i.e_ there is
no direct measurement of the frequency ratios between the additional
transition and some of the initial $n$ transitions. To account for this, we
may choose a different weight for the additional transition, say $w_{n+1}=\eta
w_{k}$, $1\leq k\leq n$. The addition of a partially connected transition to
the initial pool of fully connected transition may result in an unwanted
deviation from optimality in the realisation of the unit with the initial
transitions, that we aim at quantifying.

The computation of the uncertainty $\delta N_{i}/N_{i}$ is reported in A. For
completely correlated frequency ratio measurements, the addition of the new
transition has the same effect as adding a fully connected transition, that is
to say a marginal added uncertainty. For uncorrelated frequency ratio
measurements, the uncertainty is plotted in Figure 2 as a function of $\eta$
and $n$. It shows that for a pool of several transitions, the uncertainty of
the realisation of the unit is not significantly altered by adding a partially
connected transition to the unit, and that the degradation can be mitigated by
setting a lower weight $\eta$ to the added transition. This allows to
integrate the newly added transition into the unit, with an uncertainty on the
realisation of the unit by this transition that is a decreasing function of
$\eta$ and of the number of measured frequency ratios with the initial
transitions.

###  7.3 Two different clock transitions

Figure 3: Frequency unit defined from a set of two transitions. The graph
shows the relative uncertainty with which the unit can be realised with a
clock based on transition 1, in units of $u_{1}$, as a function of
$\eta=u_{2}/u_{1}$. The various curves represent different weighting functions
$w_{1}=\eta^{l}w_{2}$.

We now consider the case in which the unit is composed of two clock
transitions 1 and 2. We note $\eta=u_{2}/u_{1}$. The realisation of the
frequency unit with a clock based on transition 1 reads:

$\left.\frac{\delta\nu}{\nu}\right|_{1}=\sqrt{1+w_{2}^{2}(1+\eta^{2})}\
u_{1}.$ (16)

We now have to decide on the weight to attribute to each clock transition. A
first choice would be to set $w_{1}=\eta w_{2}$. However, this choice is such
that the uncertainty on the realisation of the frequency unit with a clock
based on transition 1 given by equation (16) is minimal for $\eta=1$, at
$\sqrt{3/2}\,u_{1}$, and then increases to $\sqrt{2}\,u_{1}$ for $\eta$ going
to infinity. This situation is not suitable, as adding a clock transition with
a large uncertainty would degrade the ability to realise the frequency unit
with a low uncertainty clock. To remedy this problem, one can choose
$w_{1}=\eta^{l}w_{2}$ with $l>1$. In such a case,
$\left.\delta\nu/\nu\right|_{1}$ goes to $u_{1}$ for $\eta\gg 1$ and to
$\sqrt{2}\,u_{1}$ for $\eta\ll 1$. Now, the uncertainty of the realisation of
the unit with the best clock is only limited by the uncertainty of this clock,
while realising the unit with the worse clock is $\sqrt{2}$ the clock
uncertainty (one uncertainty to for the connection to the unit _i.e._
measuring the frequency ratio, one uncertainty for the realisation). These
results are represented in figure 3.

##  8 Evolution

In the previous section, we showed that the weight of each transition in the
definition of the unit should be adapted to the uncertainty of this
transition. As the performances of the clocks improve, and as new transitions
are evaluated, we may consider altering the weighting coefficients $w_{i}$. In
doing so, the unit must be backward compatible. For this, an updated value of
the normalisation constant $N$ must be calculated from the evolution of the
weights and the transitive frequency ratio matrix $\bar{\rho}$ available from
measurements.

We denote with a superscript (m) the value of the defining constants $N$ and
$w_{k}$ at the $m$th evolution of the unit. Equating the realisation of the
unit for an arbitrary transition $i$, given by equation (12), before and after
the change yields:

$N^{(m+1)}=N^{(m)}\
\frac{\prod_{k\in\mathcal{C}}\bar{\rho}_{ik}^{w_{k}^{(m)}}}{\prod_{k\in\mathcal{C}}\bar{\rho}_{ik}^{w_{k}^{(m+1)}}}.$
(17)

Using the transitivity property of the matrix $\bar{\rho}$, it is
straightforward to check that $N^{(m+1)}$ does not depend on the choice of
$i$.

Because the values of the measured frequency ratios $\bar{\rho}$ contain
statistical noise, Equation (17) can be interpreted as a random walk of the
logarithm of $N$. As such, the unit may drift away from a previous definition
more than the uncertainty of its realisations. However, according to
Kolmogorov’s three-series theorem, the random walk converges if the series

$\sum_{m=0}^{\infty}\ v_{m}\qquad\textrm{with}\quad
v_{m}=\sum_{k\in\mathcal{C}}\
\left[u_{k}^{(m+1)}\left(w_{k}^{(m+1)}-w_{k}^{(m)}\right)\right]^{2}$ (18)

converges over $m$. This condition is easily met if the unit is updated when
the terms $v_{m}$ go to zero faster than $1/m$, and _a fortiori_ if they
improve by a fixed factor between each update of the unit. The expression of
$v_{m}$ in equation (18) is valid for a complete set of fully correlated
frequency ratio measurements, but equation (28) can be used to compute $v_{m}$
for any given covariance matrix resulting from the fit of a set of frequency
ratio measurements.

In order to illustrate the convergence of the unit, we conduct a numerical
simulation of its evolution. For this, we fix exact frequency ratios. At each
evolution step $m$, we draw random frequency ratios $\rho^{\textrm{\tiny m}}$
that deviate from the exact ratios by the uncertainty set for each transition
at step $m$. From these ratios, we estimate the most probable transitive
frequency ratio matrix $\bar{\rho}$ with a least square method. We then use
this ratio matrix to update the normalisation constant of the unit $N$
according to equation (17). Equipped with the knowledge of the exact frequency
ratios that our simulation provides us with, we then calculate the frequency
of the unit and the accuracy of its realisation with clocks based on each
transition using equations (11) and (12) respectively.

Figure 4: Simulation of the evolution of the frequency unit, starting with an
initial configuration ($m=0$) in which a single transition with an uncertainty
$u_{1}=2\times 10^{-16}$ contributes to the unit, _i.e_ reproducing the
current SI second. At the first evolution step ($m=1$), we start incorporating
another microwave clock transition (labeled 2) and three optical clock
transitions (labeled 3 to 5) to the unit. Each clock transition contributes to
the unit with a weight $w_{i}\propto 1/u_{i}^{2}$. We model a possible
stochastic evolution of the transitions’ uncertainty with only marginal
improvement for the microwave transitions, and different rates of improvement
for the three optical transitions.
Bottom right: Example trajectories of the unit, with respect to its initial
definition, corresponding to different measurements of frequency ratios. At
$m=1$, the unit experiences a step after the introduction of the new
transitions. However, this step lies withing the uncertainty of the first
transition initially defining the unit. After this first step, the unit
converges fast to an asymptotic value, noted $\nu^{(\infty)}$ and represented
for a specific highlighted trajectory.
The mains graphs show statistics over a large number of trajectories, for
three different configurations in which all the frequency ratios between all
transitions have been measured. For $a)$ the measurements are independent and
limited by statistical noise; For $b)$ and $c)$ the measurements are maximally
correlated (_i.e._ limited by common systematic effects, or simultaneous); For
$c)$, a sixth clock transition (labeled 6) with a radically improved
uncertainty is introduced after step $m=7$. For all graphs, the thin lines
show the transition uncertainties $u_{i}$ chosen for the simulation. The thick
lines show the relative standard deviation in the realisation of the unit with
a clock based on transition $i$, _i.e_ $\left.\delta\nu/\nu\right|_{i}$,
matching its analytical expression given by equation (14). The colored area
between the thin and thick line thus represents the overhead due to the non-
optimality of the unit. Figure 5:  (continued caption of Figure 4) For the
initial configuration $m=0$, this area is null for the unique primary
frequency standard (PFS, $i=1$) defining the unit, and large for the other
transitions acting as SRS. In case $b)$ and $c)$, even though the uncertainty
of the SRS is better than the uncertainty of the PFS, the former can only
realise the unit with an uncertainty limited by the accuracy of the latter, as
one can expect. However, in case $a)$ the statistical independence between the
frequency ratio measurements enables a better realisation of the unit with
SRS.
As soon as more transitions are introduced ($m>0$), the thickness of the area
is small compared to the respective transition uncertainty, meaning that it is
possible to realise the unit with a clock based on any transition with an
uncertainty very close to the nominal uncertainty of the clock. For $c)$, the
newly added transition with a much better uncertainty becomes _de facto_ the
new PFS, and the overhead in realising the unit with the other clock
transitions increases.
The thick purple line shows the relative standard deviation of the unit with
respect to its asymptotic value output from the simulation; it matches its
analytical expression
$\left(\sum_{m^{\prime}=m}^{\infty}v_{m^{\prime}}\right)^{1/2}$ where
$v_{m^{\prime}}$ is given by equation (28). In the correlated cases $b)$ and
$c)$, $v_{m^{\prime}}$ reduces to the expression of equation (18). From this
equation, we expect that if the weights $w_{i}$ smoothly evolve, _i.e._
$|w_{k}^{(m+1)}-w_{k}^{(m)}|\ll 1$, the change in the unit can be
significantly lower than the uncertainties of the transitions $u_{i}$. This
effectively appears in the simulations $a)$ and $b)$. In $c)$ however, the
abrupt change in the weights due to the sudden introduction of the additional
transition brings the change in the unit closer to the uncertainty of the
transitions.

In a first simulation, we keep the transition uncertainties constant, but the
weight of each transition is randomly redistributed at each update. This
simulation shows the expected random walk behavior of the unit as the series
(18) does not converge. In a second simulation, we show the convergence of the
unit when the transition uncertainties are randomly improved at each update of
the unit (Figure 4).

##  9 From a single primary frequency standard to multiple clock transitions

Let us consider the special case for which $w_{i_{0}}=1$ for a specific clock
transition $i_{0}$, and $w_{i}=0$ for all other transitions. In this case, the
$N_{i}$ factors reduce to:

$N_{i_{0}}=N\qquad\textrm{and}\qquad
N_{i}=N\bar{\rho}_{ii_{0}}\quad\textrm{for }i\neq i_{0},$ (19)

such that the frequency unit can be directly realised with a clock based on
the transition $i_{0}$ without additional uncertainty beside the uncertainty
of the clock, while it can be realised with clocks based on other transitions
with the added uncertainty on the frequency ratio $\bar{\rho}_{ii_{0}}$. This
situation is precisely the current situation, in which we have a primary
frequency standard (PFS) $i_{0}$ with which the frequency unit is defined by
$\nu=\nu_{i_{0}}/N_{i_{0}}=\nu_{i_{0}}/N$ exactly, and secondary
representations of the second (SRS) whose recommended frequency is
$\nu_{i}^{\textrm{\tiny SRS}}=N_{i}\nu=N\bar{\rho}_{ii_{0}}\nu$. In that view,
the unit proposed in this paper is not an alternative to the current
definition of the SI second with a single PFS and a set of SRS, but rather a
generalisation thereof. This distinction has two important consequences.

First, if at some point in the future, a specific clock transition $i_{0}$
stands out with realisations significantly more accurate than the realisations
of other transitions, the unit will evolve, according to the procedure
described in the previous section, to the weights $w_{i}=\delta_{i,i_{0}}$.
The unit would thus effectively fall back to a definition based on a single
PFS, while the other transitions would acquire the status of the current SRS.
However, this situation is reversible, should the other transitions catch up
with the transition $i_{0}$.

Second, all the algorithms or procedures already in place designed to make use
of data produced by primary and secondary standards can be readily adapted to
the unit proposed here, by replacing the recommended frequency
$\nu^{\textrm{\tiny SRS}}=N^{\textrm{\tiny SI}}\bar{\rho}_{i,\textrm{\tiny
Cs}}$ and its uncertainty for SRS by $N_{i}$ and its uncertainty for all
transitions. A notable example is the steering of TAI (Temps Atomique
International) by contributing atomic clocks. While TAI was historically
steered by the 133Cs primary standards, contributions from 87Rb, and more
recently from optical clocks (based on neutral 87Sr and 171Yb) are now
incorporated in TAI, using the recommended frequency for these SRS, and adding
its uncertainty to the clock uncertainty [11, 12]. With the new unit proposed
in this paper, all contributing clocks would be steering as SRS are steering
now. Namely, one would produce the frequency unit by scaling the frequency of
a clock based on the atomic transition $i$ by the factor $1/N_{i}$, as
expressed by equation (12), and use it to calibrate the frequency of the local
oscillator used as a pivot to connect to TAI. The relative uncertainty on
$N_{i}$ would be used to fill the $u_{\tiny S\textrm{rep}}$ entry of the
circular T. With the definition of the time unit proposed here, TAI would thus
remain a truly atomic time, in the sense that all atomic transitions would
contribute to TAI on an equal footing, with an added uncertainty $\delta
N_{i}/N_{i}$ expressing how well the transition is connected to the unit.

##  10 Practical realisation

The practical definition of the unit requires a designated body in charge of
deciding of the list of clock transitions contributing to the unit and their
respective weights. The Frequency Standards Working Group (WGFS) of the CCL
(Comité Consultatif des Longueurs) - CCTF has defined a set of requirements
and guidelines for the inclusion of transitions and frequency ratio
measurements to obtain the list of recommended frequencies, the best of which
being promoted to SRS [7]. It takes into account published frequency ratio
measurements and sets an uncertainty for these ratios that depends on the
published uncertainty, but also on the level of confidence provided by the
repeatability of the measurements by different laboratories, and possible
correlations between the frequency ratios. These conservative guidelines
ensure that the set of SRS is not affected by possible outliers.

This very procedure can be used as is to set the weight of each transition
that compose the unit proposed in this paper, and therefore to set the
definition of the unit. In addition to setting the definition itself, the
working group could recommend values for the quantities $N_{i}$ and their
uncertainty $\delta N_{i}$ required to realise the frequency unit with a clock
based on transition $i$. The former can already be computed from the published
values of the recommended frequencies for SRS, while the latter can be deduced
from the covariance matrix of the adjusted frequency ratios, a by-product of
the fitting procedure. Using these numerical constants, any laboratory
equipped with a clock based on one of the transitions composing the unit would
be able to realise it.

The shift to the unit proposed in this paper would thus comprise the following
steps:

  1. 1.

The definition of the SI second is changed to:

> The second, symbol s, is the SI unit of time. It is defined by taking the
> fixed numerical value of the weighted geometric mean of the frequencies of a
> set of atomic clocks transitions, to be a constant $N$ when expressed in the
> unit Hz, which is equal to s-1. The weights and the constant $N$ are
> published by the CIPM and updated according to frequency ratio measurements,
> in order that the unit converges.

  2. 2.

At each meeting of the CCTF, a transitive list of recommended frequency ratios
and their covariance matrix is published, along with the associated $N_{i}$
factors and their uncertainties. These quantities are derived from a fit of
all published frequency ratio measurements [8, 9]. As it is currently done for
the recommended frequencies of SRS, additional safeguards can be applied in
order to prevent outliers, such a requiring that frequency ratios should be
directly or indirectly measured by at least two independent sets of clocks, or
adding a margin to the published uncertainties.

  3. 3.

At each meeting of the CCTF, the possibility to update the weights of the
clock transitions is evaluated. It could for instance be decided each time the
quantity $v_{m}$ is reduced by a constant factor, such as two. This results in
an exponential decay of the $v_{m}$ terms with $m$, thus ensuring the rapid
convergence of their series. Such an update would notably occur at the first
meeting, so that the weights and the normalisation constant would depart from
their current value $w_{i}=\delta_{i,\textrm{\tiny Cs}}$ and
$N=9\,192\,631\,770$.

The unit proposed in this paper is particularly relevant when a large number
of frequency ratios between optical clock transitions have been measured,
hence bringing it closer to optimality. The availability of such frequency
ratios is now becoming a reality with optical clocks being compared through
optical fiber links, such that changing the definition of the unit could be
foreseen in the next few years. Yet, in order to illustrate the evolution
process of the unit, the B shows how the unit would be now defined if it had
been adopted before the 2015 meeting of the CCTF at which a few optical-to-
optical frequency ratios were considered for the determination of the
recommended frequencies of SRS.

The evolution step (iii) makes that the effective definition of the unit is
regularly updated, by adopting new weights. One could thus argue that, through
this process, the unit is constantly changing, making it indecisive. However,
in essence, the unit is not changing: it keeps being the value $v^{(\infty)}$
whose existence is ensured by the convergence criterion that must be met, and
each evolution step brings us closer to begin able to realise it, as
illustrated by example trajectories of the unit shown on figure 4. Admittedly,
$v^{(\infty)}$ is accidental because it depends on the outcome of the
frequency ratio measurements that drive the evolution of the unit. However, it
is the case for all base units of the SI: the chosen values of fundamental
constants are experimental accidents, but become defining once fixed.

##  11 Are we fixing frequency ratios?

The new SI adopted by the 26th CGPM emphasizes on the essence of fixing
physical constants to define base units. For instance, the meter is defined by
fixing the speed of light, the kilogram by fixing the Planck constant,…. In
this new SI, the wording of the definition of the SI second has been altered
in order to make explicit that the unit of time is defined by fixing the
frequency of the Cs ground state hfs to $N^{\textrm{\tiny SI}}$, as reported
in the introduction. Therefore, the question arises of which physical constant
is fixed in the frequency unit proposed in this paper. One would be tempted to
answer that frequency ratios are fixed because agreed-upon values of these
ratios are necessary in order to realise the unit using equation (12). Such a
statement would be contradictory, because these frequency ratios are
dimension-less, measurable quantities imposed to us by Nature, that we are
thus not allowed to arbitrarily fix.

Indeed, it is crucial to notice that the _definition_ itself, given by
equation (11) does not rely of the availability nor on the value of these
ratios, but only on the arbitrarily fixable constants $N$ and $w_{i}$. Only
the _realisation_ of the unit requires the knowledge of the frequency ratios.
But it is already the case that measurable physical quantities are required in
order to realise the SI second. For instance, the polarisability of the Cs
atom is required to correct for the black-body radiation shift, the quadratic
Zeeman shift coefficient is required to cancel this effect, the interaction
strength between Cs atoms is required to compensate for collisional effects,
…Without prior measurements of these measurable physical quantities, it would
not be possible to realise the SI second with Cs clocks. But the value of the
Cs polarisability itself, although required by the definition, is not stated
nor mentioned in the definition of the SI second, which is above such
mundanes. In the same spirit, the prior knowledge of measurable frequency
ratios is required to realise the unit proposed in this paper, but they are
not part of the definition, and _a fortiori_ not fixed by the definition.
However, the availability, the uncertainty or the reproducibility of frequency
ratios connecting a specific atomic transition to the pool of transitions
already used to define the unit may influence the decision to incorporate or
not this atomic transition to the pool. This requirement is aligned with the
roadmap for the redefinition of the SI second [7], which already requires that
a large number of frequency ratios are measured and reproduced with a low
uncertainty before a redefinition of the SI second can be agreed upon.

Unlike the other base units of the SI that are defined by directly fixing
fundamental constants such as $c$, $h$, $e$, $k_{B}$, $N_{A}$, the Cs-based
definition of the SI second only indirectly fixes a fundamental constant.
Indeed, one can write the frequency of the Cs clock transition as the product
of a constant with the dimension of a frequency, _e.g._ the Rydberg constant
$cR_{\infty}$, and a dimension-less function $\mathcal{F}_{\textrm{\tiny Cs}}$
of dimension-less physical constants that characterize the scale of the
various physical interactions at stake in the Cs atom, such as the fine
structure constant $\alpha$, the electron to proton mass ratio $m_{e}/m_{p}$,
…:

$\nu^{\textrm{\tiny SI}}=\frac{\nu_{\textrm{\tiny Cs}}}{N^{\textrm{\tiny
SI}}}=\frac{1}{N^{\textrm{\tiny SI}}}cR_{\infty}\,\mathcal{F}_{\textrm{\tiny
Cs}}(\alpha,m_{e}/m_{p},\ldots).$ (20)

Fixing the frequency of the Cs clock transition therefore amounts to fixing
$R_{\infty}$ to the value $\nu^{\textrm{\tiny SI}}N^{\textrm{\tiny
SI}}/c\mathcal{F}_{\textrm{\tiny Cs}}$, which is unknown because
$\mathcal{F}_{\textrm{\tiny Cs}}$ is not exactly calculable, and its constant
parameters only known with a finite uncertainty. While it would be
conceptually more satisfying to fix a fundamental constant such as
$cR_{\infty}$ to define the SI second, the choice of fixing the frequency of
the Cs clock transition is driven by the practical availability of clocks able
to realise the unit with the best accuracy.

This situation remains unchanged with the unit proposed in this paper,
because, even though it fixes several constants ($N$ and $w_{i}$) instead of a
single one ($\Delta\nu_{\textrm{\tiny Cs}}$), the frequency unit can still be
expressed in the form:

$\nu=\frac{1}{N}\prod_{i\in\mathcal{C}}\nu_{i}^{w_{i}}=\frac{1}{N}cR_{\infty}\,\mathcal{F}(\alpha,m_{e}/m_{p},\ldots).$
(21)

The function $\mathcal{F}$ simply becomes a combination of the different
functions $\mathcal{F}_{i}$:

$\mathcal{F}=\prod_{i\in\mathcal{C}}\mathcal{F}_{i}^{w_{i}}.$ (22)

and thus remains unknown as well. However, this expression offers new
possibilities. If the fundamental constants such as $\alpha$ are eventually
found to drift with time [13], one could envision to constraint the choice of
the weights $w_{i}$ such that $\mathcal{F}$ is insensitive to variations of
the fundamental constants. For instance, the Yb+ E3 transition has a large
negative dependence on $\alpha$ variations [13] that can be compensated for by
the positive dependence of Sr, Yb, Al+, Hg, at the expense of a reduced weight
for the Yb+ E3 transition.

##  12 Summary and conclusion

In summary, we showed that a frequency unit $\nu$ can be defined as the
weighted geometric mean of the frequencies of a set $\mathcal{C}$ of clock
transitions:

$\nu=\frac{1}{N}\prod_{i\in\mathcal{C}}\nu_{i}^{w_{i}}.$ (23)

The normalisation constant $N$ and the weights $w_{i}$ are defining constants
for the unit. The choice of the latter is driven by the relative uncertainty
$u_{i}$ of the transition with $w_{i}\propto 1/u_{i}^{2}$. The former is set
in order to ensure the continuity with a previous definition of the unit. Such
a unit is realisable with a single clock using a frequency ratio matrix
satisfying the transitivity property (1), obtained from an adjustment of a
connex but possibly incomplete set of experimentally determined frequency
ratios between the transitions. The accuracy of the realisation is smaller
than $\sqrt{2}u$ where $u$ is the uncertainty of the clock used for the
realisation, and goes to $u$ for the best clocks of the pool when the size of
the pool grows.

The backward compatible evolution of the definition of the second makes that
technical evolution in the frequency standards do not change the essence of
the definition of the frequency unit, but only the defining constants, in a
way that only improves the uncertainty of the realisation, without introducing
a drift of the unit.

Laboratories already equipped with high performance optical clocks will not be
incited to reorient their research activities to implement a clock based on an
arbitrarily chosen definition based on a single atomic species, thus avoiding
that this atomic species becomes over-represented in metrology laboratories.
On the contrary, defining the SI second with the unit proposed here would
encourage laboratories to develop pairs of clocks never demonstrated before in
order to improve the optimality of the unit. By doing so, the optical
frequency community would preserve the diversity of atomic transitions
currently being investigated, their enhanced potential, and their numerous
applications in tests of fundamental physics.

Beyond this community, the SI is a field of physics apprehensible to the
general public. It is therefore important that the choice of unit can be
easily explained with a simple wording. This necessity became recently
manifest, with the introduction of the new SI for which it is challenging to
intuitively explain the relation between the Planck constant and the kilogram.
Likewise, some aspects of the unit proposed in this paper rely on more
abstract mathematical reasoning than the current definition solely based on
the physical probing of Cs atoms. However, the complexity mainly lies in the
proofs of correctness of the unit (optimality and convergence). The basic
principle, on the other hand, is more accessible: in order to avoid an
arbitrary and ephemeral choice among many possible candidates, the second is
defined as the average frequency of the best realised atomic clocks
transitions at any time, with evidence that the unit remains consistent over
time. Such an explanation is arguably understandable while remaining correct.

## Acknowledgments

The author thanks Sébastien Bize for helpful suggestions about the manuscript.

## References

## References

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##  Appendix A Least square adjustment of frequency ratios

We consider a set $\left\\{\rho^{\textrm{\tiny m}}_{k},\ 1\leq k\leq
r\right\\}$ of measured frequency ratios between the frequencies of different
clock transitions. Because of statistical and systematic uncertainty in the
measurements, these ratios deviate from the actual frequency ratios $\rho$,
and do not fulfill the transitivity property (1). It is possible to derive a
most likely transitive frequency ratio matrix $\bar{\rho}$ from these
measurements using a least squares procedure, as detailed in [8]. For this,
one has to minimize the squared sum of the residuals:

$\chi^{2}=\Delta^{T}W^{-1}\Delta,$ (24)

where $\Delta$ is the vector of size $r$ gathering the differences
$\rho^{\textrm{\tiny m}}_{k}-\bar{\rho}_{k}$ between the measured and most
likely ratios, and $W$ is the $r\times r$ covariance matrix of the measured
ratios. Because of the transitivity relations linking the coefficients of
$\bar{\rho}$, the fitting procedure only involves $n-1$ free parameters, $n$
being the number of different clock transitions under consideration. A
convenient choice is to arbitrarily set the frequency of a given clock
transition $i_{0}$ to $\bar{\nu}_{i_{0}}=1$, and to express the frequency
ratios $\bar{\rho}_{k}$ as a function of the frequencies $\bar{\nu}_{i}$ of
the $n-1$ other transitions.

This procedure yields the most likely values for the $\bar{\nu}_{j}$, either
with a linear least squares algorithm (using a linearisation of the
frequencies $\bar{\nu}_{j}$ around an _a priori_ guess) [8] or with a non-
linear least squares algorithm (using the Jacobian of the system
$J_{kl}=\partial\Delta_{k}/\partial\bar{\nu}_{l}$). The outcome is the best
estimate for the frequency ratio matrix
$\bar{\rho}_{ij}=\bar{\nu}_{i}/\bar{\nu}_{j}$, as well as the covariance
matrix:

$\Sigma_{ij}=2\left(\frac{\partial^{2}\chi^{2}}{\partial\bar{\nu}_{i}\partial\bar{\nu}_{j}}\right)^{-1}.$
(25)

This is a square matrix with size $n-1$, but in order to simplify the
notations hereafter, we consider additional row and column at position $i_{0}$
with 0 components, giving a row and a column per clock transition. From this
covariance matrix, on can deduce the uncertainty on the frequency ratios:

$\frac{\delta\bar{\rho}_{ij}}{\bar{\rho}_{ij}}=\sqrt{\frac{\Sigma_{ii}}{{\bar{\nu}_{i}}^{2}}+\frac{\Sigma_{jj}}{{\bar{\nu}_{j}}^{2}}-2\frac{\Sigma_{ij}}{\bar{\nu}_{i}\bar{\nu}_{j}}}.$
(26)

To calculate the uncertainty on the geometric mean
$N_{i}=N\prod_{k}\bar{\rho}_{ik}^{w_{k}}$, it is convenient to choose the
fixed frequency $i_{0}=i$. In this case:

$\frac{\delta
N_{i}}{N_{i}}=\sqrt{\sum_{j,l=1}^{n}w_{j}w_{l}\frac{\Sigma_{jl}}{\bar{\nu}_{j}\bar{\nu}_{l}}}\qquad\textrm{for
}i=i_{0}.$ (27)

Similarly,

$v_{m}=\sum_{j,l=1}^{n}\left(w^{(m+1)}_{j}-w^{(m)}_{j}\right)\left(w^{(m+1)}_{l}-w^{(m)}_{l}\right)\frac{\Sigma_{jl}}{\bar{\nu}_{j}\bar{\nu}_{l}}.$
(28)

Although some steps in the calculation of
$\delta\bar{\rho}_{ij}/\bar{\rho}_{ij}$ and $\delta N_{i}/N_{i}$ depend on
$i_{0}$, these quantities themselves are eventually independent of $i_{0}$.

We now make explicit these uncertainties in a few specific cases

###  A.1 Uncorrelated complete set of measured ratios

We first assume that the measured frequency ratios form a complete matrix
(_i.e._ all possible frequency ratios between the $n$ atomic species have been
measured), and that all measurements are uncorrelated (_i.e._ the measurement
are independent, and limited by the statistical uncertainty or by systematic
effects that are not in common mode between different measurements). In this
very favourable case, there is an averaging effect between the measurements
that produce a most-likely frequency ratio matrix $\bar{\rho}$ with an
uncertainty lower than the individual uncertainty of the clocks. Explicitly,
the input covariance matrix is diagonal and reads:

$W_{kk}=\delta\rho_{ij}^{2}=\rho_{ij}^{2}\left(u_{i}^{2}+u_{j}^{2}\right),$
(29)

where $i$ and $j$ are the indexes of the two transitions involved in the
frequency ratio measurement $k$, and $u_{l}$ is the relative uncertainty of
transition $l$. Using the formulas above, one can then express the relative
uncertainty on the most likely frequency ratios as a function of the
individual transition frequencies. For instance, with a set $\mathcal{C}$ of
$n=3$ clock transitions, this uncertainty reads:

$\delta\bar{\rho}_{ij}=\sqrt{1-\frac{u_{i}^{2}+u_{j}^{2}}{2\sum_{k\in\mathcal{C}}u^{2}_{k}}}\
\delta\rho_{ij}\qquad\textrm{for }n=3.$ (30)

This equation shows that the uncertainty $\delta\bar{\rho}_{ij}$ on the most
likely frequency ratio $\bar{\rho}_{ij}$ is smaller than the uncertainty
$\delta\rho_{ij}$ on the measured frequency ratio, by exploiting the redundant
information in the uncorrelated frequency ratio matrix. For a set of $n$
transitions with the same uncertainty $u$, we have:

$\frac{\delta\bar{\rho}_{ij}}{\bar{\rho}_{ij}}=\frac{2}{\sqrt{n}}u,$ (31)

and

$\frac{\delta N_{i}}{N_{i}}=\frac{\sqrt{2(n-1)}}{n}u.$ (32)

The $1/\sqrt{n}$ behaviour of these uncertainties also illustrates the
averaging effect.

###  A.2 Fully correlated complete set of measured ratios

We now make the opposite assumption: all possible frequency ratios between the
various clock transitions are measured, but all these measurement are
performed with the same set of clocks, at the same time. This means that
statistic and systematic uncertainties are fully correlated. The diagonal
terms of the covariance matrix $W$ are unchanged, but it now features off-
diagonal terms.

For $n$ transitions with the same uncertainty $u$, this yields

$\frac{\delta\bar{\rho}_{ij}}{\bar{\rho}_{ij}}=\sqrt{2}u,$ (33)

and

$\frac{\delta N_{i}}{N_{i}}=\sqrt{1-\frac{1}{n}}u.$ (34)

The uncertainty on all frequency ratios is limited by the transition
uncertainties, and the relative uncertainty on $N_{i}$ is at most the
transition uncertainty $u$.

###  A.3 Incomplete set of measured ratios

We now detail the situation considered in section 7.2. The complete frequency
ratios are measured for a set of $n$ transitions, and a transition $n+1$, with
the same uncertainty $u$ as the initial transitions is added to the pool of
transitions, but only a limited number $n_{c}<n$ of frequency ratios involving
this last transition have been measured. We define $\eta=w_{n+1}/w_{1..n}$ the
relative weight of the additional transition in the definition of the
frequency unit. With these assumptions, the additional relative uncertainty of
the realisation of the frequency unit reads:

For uncorrelated measured frequency ratios:

$\frac{\delta
N_{i}}{N_{i}}=\frac{\sqrt{2}}{n+\eta}\sqrt{\frac{(n+n_{c})\eta^{2}+n(n_{c}-1)(2\eta-1)}{n_{c}(n+1)}+n-1}\qquad\textrm{for
}1\leq i\leq n_{c}$ (35) $\frac{\delta
N_{i}}{N_{i}}=\frac{\sqrt{2}}{n+\eta}\sqrt{\frac{n+n_{c}+1}{n\,n_{c}}\eta^{2}+2\eta+n-1}\qquad\textrm{for
}n_{c}<i\leq n$ (36) $\frac{\delta
N_{n+1}}{N_{n+1}}=\frac{\sqrt{2}}{n+\eta}\sqrt{\frac{n^{2}+n-n_{c}}{n_{c}}}$
(37)

For correlated measured frequency ratios:

$\frac{\delta
N_{i}}{N_{i}}=\frac{\sqrt{2\eta^{2}+2(n-1)\eta+n(n-1)}}{n+\eta}\qquad\textrm{for
}1\leq i\leq n$ (38) $\frac{\delta
N_{n+1}}{N_{n+1}}=\frac{\sqrt{n(n+1)}}{n+\eta}$ (39)

##  Appendix B defining the unit with the current frequency ratio measurements

2015 |  |  |
---|---|---|---
Species | $w_{i}$ | $N_{i}$ | $\delta N_{i}/N_{i}$
133Cs | 0.089 | 9192631770 | $1.7\times 10^{-16}$
171Yb | 0.005 | 518295836590865.105 | $9.4\times 10^{-16}$
171Yb+(E2) | 0.071 | 642121496772645.04 | $2.7\times 10^{-16}$
171Yb+(E3) | 0.063 | 688358979309308.379 | $8.6\times 10^{-17}$
199Hg | 0.362 | 1128575290808154.79 | $8.6\times 10^{-16}$
199Hg+ | 0.010 | 1064721609899145.3 | $6.6\times 10^{-16}$
27Al+ | 0.011 | 1121015393207857.31 | $6.4\times 10^{-16}$
87Rb | 0.020 | 6834682610.904307 | $4.7\times 10^{-16}$
87Sr | 0.368 | 429228004229873.166 | $7.4\times 10^{-17}$

2017 |  |  |
---|---|---|---
Species | $w_{i}$ | $N_{i}$ | $\delta N_{i}/N_{i}$
133Cs | 0.081 | 9192631770.00000213 | $9.5\times 10^{-17}$
171Yb | 0.327 | 518295836590863.762 | $3.8\times 10^{-17}$
171Yb+(E2) | 0.016 | 642121496772645.189 | $2.6\times 10^{-16}$
171Yb+(E3) | 0.015 | 688358979309308.539 | $2.9\times 10^{-16}$
199Hg | 0.166 | 1128575290808154.68 | $6.4\times 10^{-17}$
199Hg+ | 0.002 | 1064721609899145.55 | $6.5\times 10^{-16}$
27Al+ | 0.003 | 1121015393207857.57 | $6.3\times 10^{-16}$
87Rb | 0.037 | 6834682610.90431418 | $1.6\times 10^{-16}$
87Sr | 0.353 | 429228004229873.134 | $2.7\times 10^{-17}$

Table 2: Proposition for the weights of the various transitions composing the
unit, using the list of frequency ratio measurements published before 2015
(left) and 2017 (right), when the 20th and 21st meetings of the CCTF were
held. We calculate $\sqrt{v_{\tiny 2015}}=1.7\times 10^{-16}$ at the first
update of the unit, and $\sqrt{v_{\tiny 2017}}=3.5\times 10^{-17}$. The
significant reduction of $v$ in 2017 with respect to 2015 justifies a new
update of the unit, enabled by the inclusion of new optical frequency ratio
measurements. The normalisation constant is set to $N=203102222210226.561$ in
2015, and to $N=152329318467266.642$ in 2017, in order to ensure continuity
with the previous Cs based definition of the unit.

The BIPM website publishes the list of measured frequency ratios that is
currently being used as the input of the least-squares algorithm [8] used to
determine the recommended frequency of the SRS. This list can be used as a
playground to simulate the unit proposed in this paper, as if it had already
been adopted. From the list of frequency ratios published before a given date,
we determine the transitive set of most probable frequency ratios
$\bar{\rho}_{ij}$ and their covariance matrix $\Sigma_{ij}$.

In order to define the unit one has to choose the weights $w_{i}$. In this
paper, we prescribed to chose weights proportional to $1/u_{i}^{2}$ where
$u_{i}$ is the uncertainty of the clocks realising the transition $i$.
However, in practice, frequency ratios can be measured by different clocks
using the transition $i$ with varying statistical and systematic
uncertainties, possibly larger than the best uncertainty reported for
transition $i$. $u_{i}$ is therefore ill-defined. Instead, the relevant
uncertainty can be found in the values of $\delta\bar{\rho}_{ij}$ as given by
the fit procedure, which represents how the clock transition $i$ is connected
to other transitions by actual frequency ratio measurements. Here, we
heuristically choose $w_{i}$ to be proportional to the average of the two
largest values of $1/\delta\bar{\rho}_{ij}^{2}$ over $j$. This makes $w_{i}$
representative of the uncertainty of the best frequency ratio measurements
with transition $i$, while avoiding that a single frequency ratio would
exclusively steer the unit. One could alternatively choose the weights of the
clocks with a specific optimisation procedure in order to _e.g._ minimize the
sub-optimality of the best transitions. All in all, these possible choices
become similar as more frequency ratios between optical transitions are
measured with state-of-the-art clocks; and any of these choices is definitely
better than setting the full weight of the unit on a single atomic transition
in the current and forseable context where many different optical clocks are
developed.

From the weights $w_{i}$ and the covariance matrix $\Sigma_{ij}$, one can
calculate the factors $N_{i}$ and their uncertainties, as well as $v_{m}$ and
decide whether the unit should be updated or not. If so, the new normalisation
constant $N$ is calculated. Table 2 reports on the outcome of such a procedure
using the frequency ratio measurements published prior to 2015 and 2017.