#  A new estimate of the transfinite diameter of Bernstein sets

Dimitri Jordan Kenne  Doctoral school of Exact and Natural Sciences,
Jagiellonian University, Łojasiewicza 11, Kraków, 30-348, Lesser Poland,
Poland<EMAIL_ADDRESS>
(Date: August 28, 2024)

###### Abstract.

Let $K\subset\mathbb{C}^{n}$ be a compact set satisfying the following
Bernstein inequality: for any $m\in\\{1,...,n\\}$ and for any $n$-variate
polynomial $P$ of degree $\deg(P)$ we have

$\displaystyle\max_{z\in K}\left|\frac{\partial P}{\partial
z_{m}}(z)\right|\leq M\deg(P)\max_{z\in K}|P(z)|\ \mbox{ for
}z=(z_{1},\dots,z_{n}).$

for some constant $M=M(K)>0$ depending only on $K$. We show that the
transfinite diameter of $K$, denoted $\delta(K)$, verifies the following lower
estimate

$\displaystyle\delta(K)\geq\frac{1}{nM},$

which is optimal in the one-dimensional case. In addition, we show that if $K$
is a Cartesian product of compact planar sets then

$\displaystyle\delta(K)\geq\frac{1}{M}.$

###### Key words and phrases:

Transfinite diameter, Bernstein inequality, Markov inequality

###### 2000 Mathematics Subject Classification:

Primary 41A17, 31C15, 32U15

##  1\. Introduction

Sets verifying a Markov inequality are of particular interest in approximation
and pluripotential theories. In [8], Pleśniak posed the question of the
continuity of Green pluricomplex function for these sets. In particular, he
wanted to know whether they are nonpluripolar or not, which is equivalent to
verifying that the transfinite diameter is positive or zero (see [6]). A
subclass of these Markov sets known as Bernstein sets, have been proven by
Siciak in [9] to be nonpluripolar. Some positive lower bounds of the
transfinite diameter of Bernstein sets were proven by Białas-Cież and
Jedrzejowski (first) in [2] and Yazici (later) in [11]; therefore emphasing
their nonpluripolarity. In this paper, we present a better lower estimate of
the transfinite diameter of Bernstein sets which is optimal in the one-
dimensional case.

We consider $\mathcal{P}_{d}(\mathbb{C}^{n})$, the space of $n$-variate
polynomials of total degree at most $d$. A compact set $K\in\mathbb{C}^{n}$ is
a Markov set if it satisfies the following Markov inequality: for each
$m\in\\{1,\dots,n\\}$ and for every polynomial $P\in\mathcal{P}_{d}(C^{n})$

(1) $\max_{z\in K}\big{|}\frac{\partial P}{\partial z_{m}}(z)\big{|}\leq
Md^{r}\max_{z\in K}|P(z)|\quad\mbox{ for }z=(z_{1},\dots,z_{n})$

for some constants $M=M(K)>0$, $r=r(K)>0$ which depend only on $K$. If (1) is
verified with $r=1$ then $K$ is called a Bernstein set. For instance, the
closed unit disc $D(0,1)=\\{z\in\mathbb{C}:\ |z|\leq 1\\}$ is a well-known
Bernstein set as it satisfies the inequality

$\max_{z\in D(0,1)}\left|P^{\prime}(z)\right|\leq d\max_{z\in D(0,1)}|P(z)|$

for all polynomial $P\in\mathcal{P}_{d}(\mathbb{C})$. More generally, any
finite union of $\mathcal{C}^{2}$-smooth Jordan curves is also a Bernstein set
(see [7]). Examples of Bernstein sets with an infinite number of connected
components have been constructed in [10].

The Bernstein’s property is strongly related to the smoothness of Green
pluricomplex function. Indeed, Siciak shows in [9] that, a set is Bernstein
(or verifies a Bernstein inequality) if and only if its Green pluricomplex
function is Hölder continuous (with exponent $\mu=1$). It follows from this
equivalence that Bernstein sets are nonpluripolar. As far as we know, the
generalisation of this result to all Markov sets remains an open question (for
$n\geq 2$).

We give in the preliminary section 2, the definition of the transfinite
diameter of a given compact set $K\subset\mathbb{C}^{n}$ and throughout this
paper we denote it, $\delta(K)$. Now, we recall the lower estimates of the
transfinite diameter of Bernstein sets that have been proven so far. In [1],
Białas-Cież established that the transfinite diameter of any Markov set
$K\subset\mathbb{C}$ verifies the following estimate

(2) $\delta(K)>\dfrac{1}{M}\sigma^{r}(\mbox{diam}(K))^{\frac{1}{3}}$

where $\sigma$ is an absolute positive constant and $\mbox{diam}(K)$ is the
diameter of $K$. As a consequence, all planar Markov sets are nonpolar. Later,
Bialas-Cież and Jedrzejowski proved in [2] that any Bernstein set
$K\subset\mathbb{C}^{n}$ satisfies

(3) $\delta(K)>\dfrac{1}{M2^{n-1}}.$

Recently in [11], Yazici improved the previous lower estimate for the
transfinite diameter of any Bernstein set $K\subset\mathbb{C}^{n}$ for $n>4$,
by proving that

(4) $\delta(K)>\dfrac{1}{enM}.$

In this paper, we use an idea from [4] involving generalized Leja points to
obtain a better lower bound for the transfinite diameter of Bernstein sets.
Our main result is the following.

######  Theorem 1.1.

Let $K$ be a Bernstein set in $\mathbb{C}^{n}$ of parameter $M$. Then

(5) $\delta(K)\geq\frac{1}{nM}.$

######  Remark 1.

Note that the estimate in (5) is optimal in the one-dimensional case since the
closed disc $D(0,R)=\\{z\in\mathbb{C}:\ |z|\leq R\\}$ ($R>0$) is a Bernstein
set with $M=1/R$ and its transfinite diameter is $\delta(D(0,R))=R$. Moreover,
the estimate in (5) is better than (4) for all natural numbers $n$ and better
than (3) for all $n\geq 3$.

It turns out that the estimate given in Theorem 1.1 can be improved for the
case of a Cartesian product of planar sets.

######  Theorem 1.2.

Let $K=K_{1}\times\dots\times K_{n}$ be a Bernstein set in $\mathbb{C}^{n}$ of
parameter $M$. Then

(6) $\delta(K)\geq\frac{1}{M}.$

##  2\. Preliminaries

Let $\mathbb{N}=\\{0,1,2,\dots\\}$ be the set of natural numbers and let
$\alpha:\mathbb{N}\ni j\longmapsto(\alpha_{1}(j),\dots,\alpha_{n}(j))$ be the
enumeration on $\mathbb{N}^{n}$ associated with the graded lexicographical
ordering which we denote by “$\prec$”. We write
$|\beta|=\beta_{1}+\dots+\beta_{n}$ for the length of a multi-index
$\beta=(\beta_{1},\dots,\beta_{n})$. Consider the monomials

(7) $\displaystyle e_{j}(z):=z^{\alpha(j)}=z_{1}^{\alpha_{1}(j)}\cdots
z_{2}^{\alpha_{n}(j)},\quad j=1,2,\dots.$

The space of holomorphic polynomials of $n\geq 1$ complex variables and of
degree at most $d\in\mathbb{N}$ is

(8)
$\mathcal{P}_{d}(\mathbb{C}^{n})=\mbox{span}\\{e_{i}:=z^{\alpha(i)}=z_{1}^{\alpha_{1}(i)}\cdots
z_{n}^{\alpha_{n}(i)};\ i\in\mathbb{N}\mbox{ and }\deg(e_{i})\leq d\\}$

and its dimension is
$h_{d}:=\dim(\mathcal{P}_{d}(\mathbb{C}^{n}))=\binom{n+d}{n}$. For any set of
points $\\{\xi_{0},\dots,\xi_{k-1}\\}$ of $\mathbb{C}^{n}$, we define the
generalized Vandermonde determinant $\textsc{vdm}(\xi_{0},\dots,\xi_{k-1})$
by:

(9)
$\textsc{vdm}(\xi_{0},\dots,\xi_{k-1}):=\det[e_{i}(\xi_{j})]_{i,j=0,1,2,\dots,k-1}$

with the convention $\textsc{vdm}(\xi_{0}):=1$. For any $k\in\mathbb{N}$, we
consider the constants

(10) $V_{k}:=\max_{\xi_{0},\dots,\xi_{k-1}\in
K}\left|\textsc{vdm}(\xi_{0},\dots,\xi_{k-1})\right|,\quad
l_{d}:=\sum_{i=1}^{d}i(h_{i}-h_{i-1}).$

The constant $l_{d}$ is the total degree of
$\textsc{vdm}(\xi_{0},\dots,\xi_{h_{d}-1})$ viewed as a polynomial in
$\xi_{0},\dots,\xi_{h_{d}-1}$. It is known that $l_{d}=n\binom{n+d}{n+1}$.

The transfinite diameter of a compact set $K\subset\mathbb{C}^{n}$ is the
constant

(11) $\delta(K)=\limsup_{d\to+\infty}\delta_{d}(K),$

where $\displaystyle\delta_{d}(K)=V_{h_{d}}^{1/l_{d}}$ is called the $d$-th
order transfinite diameter of $K$. Fekete proved in [3] that the limit
$\delta(K)$ exists for any compact set $K\subset\mathbb{C}$ (i.e when $n=1$).
Later in [5], Leja introduced the name “transfinite diameter” and thus posed
the problem of its existence when $n\geq 2$. A positive answer to his problem
was given later by Zaharjuta in [12].
The transfinite diameter of a compact set can also be determined using Leja
sequences. A Leja sequence for a compact set $K\subset\mathbb{C}^{n}$ is a
sequence $(\xi_{j})_{j\geq 0}$ such that $\xi_{0}$ is any arbitrary point
(preferably on $\partial K$) and for each $N\geq 1$

(12) $|\textsc{vdm}(\xi_{0},\dots,\xi_{N-1},\xi_{N})|=\sup_{z\in
K}|\textsc{vdm}(\xi_{0},\dots,\xi_{N-1},z)|.$

It is proved in [4] that, for any compact set $K\subset\mathbb{C}^{n}$

(13)
$\delta(K)=\lim_{d\to+\infty}|\textsc{vdm}(\xi_{0},\dots,\xi_{h_{d}-1})|^{1/l_{d}}$

We Consider the following mappings

(16)
$\displaystyle\begin{array}[]{rccl}\partial_{i}:&\mathbb{N}^{n}&\longrightarrow&\mathbb{N}^{n}\\\
&\beta&\longmapsto&\partial_{i}\beta=\begin{cases}(0,\dots,0)&\text{if
}\beta_{i}=0\\\
(\beta_{1},\dots,\beta_{i-1},\beta_{i}-1,\beta_{i+1},\dots,\beta_{n})&\text{if
}\beta_{i}\geq 1\end{cases},\end{array}$

$i=1,\dots,n$.

Observe that $\displaystyle\dfrac{\partial z^{\beta}}{\partial
z_{m}}=\beta_{m}z^{\partial_{m}(\beta)}$ for any multi-index
$\beta\in\mathbb{N}^{n}$ and for any $i=1,\dots,n$.
The following lemma is a direct consequence to the fact that the graded
lexicographical order is translation invariant.

######  Lemma 2.1.

Let $\beta=(\beta_{1},\dots,\beta_{n})$ and
$\gamma=(\gamma_{1},\dots,\gamma_{n})$ be two multi-indices in
$\mathbb{N}^{n}$. If $\beta\prec\gamma$ then for any $i\in\\{1,\dots,n\\}$
such that $\gamma_{i}\geq 1$ we have
$\partial_{i}\beta\prec\partial_{i}\gamma$ or
$\partial_{i}\beta=\partial_{i}\gamma=(0,\dots,0)$.

###### Proof.

Let $\beta\prec\gamma$ and let $i\in\\{1,\dots,n\\}$ such that $\gamma_{i}\geq
1$. We distinguishes two cases.

  * •

If $\beta_{i}=0$ then $\partial_{i}\beta=(0,\dots,0)$. Hence,
$\partial_{i}\beta=\partial_{i}\gamma=(0,\dots,0)$ or
$\partial_{i}\beta\prec\partial_{i}\gamma$.

  * •

Suppose that $\beta_{i}\geq 1$. We have
$\partial_{i}\beta=(\beta_{1},\dots,\beta_{i-1},\beta_{i}-1,\beta_{i+1},\dots,\beta_{n})$
and
$\partial_{i}\gamma=(\gamma_{1},\dots,\gamma_{i-1},\gamma_{i}-1,\gamma_{i+1},\dots,\gamma_{n})$.
There are two cases to consider:

    * –

If $|\beta|<|\gamma|$ then
$|\partial_{i}\beta|=|\beta|-1<|\gamma|-1=|\partial_{i}\gamma|$. It follows
directly from the definition of the graded lexicographical order that
$\partial_{i}\beta\prec\partial_{i}\gamma$.

    * –

If $|\beta|=|\gamma|$ then we also have
$\partial_{i}\beta\prec\partial_{i}\gamma$ since the lexicographical order is
translation invariant and by hypothesis $\beta\prec\gamma$.

∎

##  3\. Proofs of Theorem 1.1 and Theorem 1.2

We begin by establishing two lemmas (Lemma 3.1 and Lemma 3.2) which are
obtained by using the Markov’s property over the following class of
polynomials

(17) $\mathcal{P}^{i}=\\{P_{i}(z)=e_{i}(z)+\sum_{0\leq j<i}c_{j}e_{j}(z):\
c_{j}\in\mathbb{C}\\}\quad\text{for }i\in\mathbb{N}.$

It is essentially due to these results that we can obtain a better estimate of
the transfinite diameter of Bernstein sets.

######  Lemma 3.1.

Let $K$ be a Markov set in $\mathbb{C}^{n}$ of parameters $(M,r)$. For every
polynomial
$P_{i}(z)=e_{i}(z)+\sum_{0\leq j<i}c_{j}e_{j}(z)\in\mathcal{P}^{i}$, we have

(18) $\max_{z\in K}|D^{\alpha(i)}P_{i}(z)|\leq
M^{|\alpha(i)|}(|\alpha(i)|!)^{r}\ \max_{z\in K}|P_{i}(z)|$

where $D^{\alpha}P:=\dfrac{\partial^{\absolutevalue{\alpha}}P}{\partial
z_{1}^{\alpha_{1}}\cdots\partial z_{n}^{\alpha_{n}}}$.

###### Proof.

We give a proof by induction on $i\in\mathbb{N}$. For $i=0$ the inequality
(18) is easily verified since we have
$D^{\alpha(0)}P_{0}=D^{(0,\dots,0)}(1)=1$. Inductive step: fix $k\geq 1$ and
suppose that (18) is true for all $0\leq i\leq k-1$. Let us show that it is
also verified at the rank $i=k$. Let $d\geq 0$ such that $h_{d}\leq
k<h_{d+1}$. Since $|\alpha(k)|\geq 1$, we can select a certain
$m\in\\{1,\dots,n\\}$ for which $\alpha_{m}(k)$ is positive. Let
$k_{0}\in\\{0,\dots,k\\}$ such that

$\alpha(k_{0})=\partial_{m}\alpha(k)=(\alpha_{1}(k),\dots,\alpha_{m-1}(k),\alpha_{m}(k)-1,\alpha_{m+1}(k),\dots,\alpha_{n}(k)).$

We have $D^{\alpha(k)}P_{k}=D^{\alpha(k_{0})}\left(\frac{\partial
P_{k}}{\partial z_{m}}\right)$. Moreover, Lemma 2.1 guarantees that we can
write

$\frac{\partial P_{k}}{\partial
z_{m}}=\alpha_{m}(k)z^{\alpha(k_{0})}+\sum_{j=0}^{k_{0}}b_{j}e_{j}=\alpha_{m}(k)\left(e_{k_{0}}+\sum_{j=0}^{k_{0}}\tilde{c}_{j}e_{j}\right).$

for some constants $b_{j},\tilde{c}_{j}\geq 0$ (eventually equal to zero),
$j=1,\dots,k_{0}$.
By setting $Q_{k_{0}}=e_{k_{0}}+\sum_{j=0}^{k_{0}}\tilde{c}_{j}e_{j}$, it
follows that $D^{\alpha(k)}P_{k}=\alpha_{m}(k)D^{\alpha(k_{0})}Q_{k_{0}}$.
Using the hypothesis of induction with the polynomial $Q_{k_{0}}$, we deduce
that

$\displaystyle\max_{z\in K}|D^{\alpha(k)}P_{k}(z)|$
$\displaystyle=\alpha_{m}(k)\max_{z\in
K}|D^{\alpha(k_{0})}Q_{k_{0}}(z)|\leq\alpha_{m}(k)\
M^{|\alpha(k_{0})|}(|\alpha(k_{0})|!)^{r}\ \max_{z\in K}|Q_{k_{0}}(z)|$
$\displaystyle\leq\left(\alpha_{m}(k)\
M^{|\alpha(k_{0})|}(|\alpha(k_{0})|!)^{r}\right)\left(\frac{1}{\alpha_{m}(k)}\max_{z\in
K}\left|\frac{\partial P_{k}}{\partial z_{m}}(z)\right|\right)$
$\displaystyle=M^{|\alpha(k_{0})|}(|\alpha(k_{0})|!)^{r}\ \max_{z\in
K}\left|\frac{\partial P_{k}}{\partial z_{m}}(z)\right|.$

Moreover, from the definition of Markov set we have

$\max_{z\in K}\left|\frac{\partial P_{k}}{\partial z_{m}}(z)\right|\leq
M|\alpha(k)|^{r}\max_{z\in K}|P_{k}(z)|.$

Therefore,

$\max_{z\in K}|D^{\alpha(k)}P_{k}(z)|\leq
M^{|\alpha(k_{0})|+1}|\alpha(k)|^{r}(|\alpha(k_{0})|!)^{r}\ \max_{z\in
K}|P_{k}(z)|\leq M^{|\alpha(k)|}(|\alpha(k)|!)^{r}\ \max_{z\in K}|P_{k}(z)|$

since $|\alpha(k)|=|\alpha(k_{0})|+1$. Thus (18) is true for all $i\geq 0$. ∎

The following Lemma provides a better version of Inequality (18) for the case
of Cartesian product sets.

######  Lemma 3.2.

Let $K=K_{1}\times\dots\times K_{n}$ be a Markov set in $\mathbb{C}^{n}$ of
parameters $(M,r)$. For every polynomial $P_{i}(z)=e_{i}(z)+\sum_{0\leq
j<i}c_{j}e_{j}(z)\in\mathcal{P}^{i}$, we have

(19) $\max_{z\in K}|D^{\alpha(i)}P_{i}(z)|\leq
M^{|\alpha(i)|}(\alpha(i)!)^{r}\ \max_{z\in K}|P_{i}(z)|,$

where $\alpha!=\alpha_{1}!\cdots\alpha_{n}!$.

###### Proof.

The proof is similar to that of Lemma 2.1. We proceed by induction on
$i\in\mathbb{N}$. The case $i=0$ is obvious. Now, fix $k\geq 1$,
$P_{k}\in\mathcal{P}^{k}$ and suppose that the inequality (19) is true for all
$0\leq i\leq k-1$. We can assume without loss of the generality that
$\alpha_{1}(k)\geq 1$. Let us consider $k_{0}\in\\{0,\dots,k-1\\}$ such that
$\alpha(k_{0})=\partial_{1}\alpha(k)$. By setting
$Q_{k_{0}}=e_{k_{0}}+\sum_{j=0}^{k_{0}}\tilde{c}_{j}e_{j}$ so that
$D^{\alpha(k)}P_{k}=\alpha_{1}(k)D^{\alpha(k_{0})}Q_{k_{0}}$, as in the proof
of Lemma 3.1, we obtain by the hypothesis of induction

(20) $\max_{z\in K}|D^{\alpha(k)}P_{k}(z)|\leq
M^{|\alpha(k_{0})|}(\alpha(k_{0})!)^{r}\ \max_{z\in K}\left|\frac{\partial
P_{k}}{\partial z_{1}}(z)\right|.$

For $\xi_{2},\dots,\xi_{n}\in\mathbb{C}$ fixed, the polynomial
$P_{k}(z_{1},\xi_{2},\dots,\xi_{n})$ belongs to
$\mathcal{P}_{\alpha_{1}(k)}(\mathbb{C})\subset\mathcal{P}_{\alpha_{1}(k)}(\mathbb{C}^{n})$.
Therefore, the Markov inequality gives

$\left|\frac{\partial P_{k}}{\partial
z_{1}}(z_{1},\xi_{2},\dots,\xi_{n})\right|\leq M(\alpha_{1}(k))^{r}\max_{z\in
K}|P_{k}(z)|\quad\text{for all }z_{1}\in K_{1}.$

Since $\xi_{2},\dots,\xi_{n}\in\mathbb{C}$ are arbitrarily chosen, it follows
that

$\max_{z\in K}\left|\frac{\partial P_{k}}{\partial z_{1}}(z)\right|\leq
M(\alpha_{1}(k))^{r}\max_{z\in K}|P_{k}(z)|$

and hence the inequality (20) implies

$\displaystyle\max_{z\in K}|D^{\alpha(k)}P_{k}(z)|$ $\displaystyle\leq
M^{\absolutevalue{\alpha(k_{0})}}(\alpha(k_{0})!)^{r}\
\left(M\alpha_{1}(k)^{r}\max_{z\in K}|P_{k}(z)|\right)$
$\displaystyle=M^{|\alpha(k)|}(\alpha(k)!)^{r}\ \max_{z\in K}|P_{k}(z)|.$

because $\absolutevalue{\alpha(k_{0})}+1=\absolutevalue{\alpha(k)}$ and
$\alpha(k_{0})!=(\alpha_{1}(k)-1)!\alpha_{2}(k)!\cdots\alpha_{n}(k)!$. ∎

######  Remark 2.

We deduce from Lemma 3.1 and Lemma 3.2 that the inequalities (18) and (19) are
valid for all polynomials $P_{i}(z)=\sum_{j=0}^{i}c_{j}e_{j}(z)$, with
$c_{j}\in\mathbb{C}$.

######  Remark 3.

It is interesting to note that we have the following result: for all compact
sets $K_{j}\subset\mathbb{C}^{n_{j}}$ ($n_{j}\in\mathbb{N}$), $j=1,\dots,m$

(21) $K_{1}\times\cdots\times K_{m}\subset\mathbb{C}^{n}\mbox{ is a Markov set
}\Longleftrightarrow K_{1},\dots,K_{m}\ \mbox{ are all Markov sets}.$

Indeed, if $K_{1}\times\cdots\times K_{m}$ is a Markov set of parameters
$(M,r)$ then using the fact that for each $d\in\mathbb{N}$ and for each
$j=1,\dots,m$,
$\mathcal{P}_{d}(\mathbb{C}^{n_{j}})\subset\mathcal{P}_{d}(\mathbb{C}^{n})$,
we can easily deduce that $K_{1},\dots,K_{m}$ are also Markov sets of
parameters $(M,r)$. On the other hand, if we suppose that $K_{1},\dots,K_{m}$
are Markov sets with respective parameters
$(M_{1},r_{1}),\dots,(M_{m},r_{m})$, it is straightforward to see that
$K_{1}\times\cdots\times K_{m}$ is a Markov set of parameters
($\displaystyle\max_{j=1,\dots,m}M_{j},\max_{j=1,\dots,m}r_{j}$).

Now we can prove Theorem 1.1 and Theorem 1.2.

###### Proof of Theorem 1.1.

Let $K$ be a Bernstein set in $\mathbb{C}^{n}$ and let $(\xi_{i})_{i\geq 1}$
be a Leja sequence for $K$. Then $K$ is necessarily determining for the space
of polynomial as a Bernstein set, i.e $P\in\bigcup_{d\geq
0}\mathcal{P}_{d}(\mathbb{C}^{n})$ and $P\equiv 0$ on $K$ imply $P\equiv 0$ in
$\mathbb{C}^{n}$. Therefore, all Leja sequences for $K$ are unisolvent, i.e

$\textsc{vdm}^{(n)}(\xi_{0},\dots,\xi_{i-1})\not=0\quad\mbox{for all $i\geq
1$}.$

Set

$P_{i}(\xi):=\dfrac{\textsc{vdm}(\xi_{0},\dots,\xi_{i-1},\xi)}{\textsc{vdm}(\xi_{0},\dots,\xi_{i-1})}\quad\mbox{
for all $i\geq 1$}.$

By expanding the Vandermonde determinant, which is in the numerator of
$P_{i}$, following its last column we obtain for each $i\geq 1$

$P_{i}(\xi)=e_{i}(\xi)+\sum_{0\leq j<i}c_{j}e_{j}(\xi).$

for some constants $c_{i}\in\mathbb{C}$. Now observe that
$\textsc{vdm}(\xi_{0},\dots,\xi_{i-1},\xi_{i})=\prod_{j=1}^{i}P_{j}(\xi_{j})$,
$i\geq 1$. Hence, using the definition of Leja sequence and then Lemma 3.1 we
obtain

$\displaystyle|\textsc{vdm}(\xi_{0},\dots,\xi_{h_{d}-1})|$
$\displaystyle=\left|\prod_{i=1}^{h_{d}-1}P_{i}(\xi_{i})\right|=\prod_{i=1}^{h_{d}-1}\max_{z\in
K}\absolutevalue{P_{i}(z)}$
$\displaystyle\geq\prod_{i=1}^{h_{d}-1}\left[\frac{1}{M^{|\alpha(i)|}|\alpha(i)|!}\norm{D^{\alpha(i)}P_{i}}_{K}\right]=\prod_{i=1}^{h_{d}-1}\left[\frac{\alpha(i)!}{M^{|\alpha(i)|}|\alpha(i)|!}\right]$
$\displaystyle\geq\prod_{i=1}^{h_{d}-1}\left[\frac{1}{(nM)^{|\alpha(i)|}}\right]=\frac{1}{(nM)^{l_{d}}}.$

The last inequality is due to the fact that
$D^{\alpha(i)}P_{i}=\alpha(i)!\geq\frac{1}{n^{|\alpha(i)|}}|\alpha(i)|!$ and
$l_{d}=\sum_{i=1}^{h_{d}-1}|\alpha(i)|$. It follows that

$\delta(K)=\lim_{d\to+\infty}|\textsc{vdm}(\xi_{0},\dots,\xi_{h_{d}-1})|^{1/l_{d}}\geq\frac{1}{nM}$

which is the desired estimate. ∎

###### Proof of Theorem 1.2.

The proof is the same as that of Theorem 1.1 but here we use Lemma 3.2 to
replace Lemma 3.1. ∎

## Acknowledgments

I wish to thank Professor Leokadia Białas-Cież for her valuable comments on
this work.
Funding: This work was partially supported by the National Science Center,
Poland, grant Preludium Bis 1 No 2019/35/O/ST1/02245

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# Absolute frequency measurement of the 87Sr optical lattice clock at NTSC
using International Atomic Time

Xiaotong Lu1, Feng Guo1,2, Yebing Wang1,2, Qinfang Xu1,2, Chihua Zhou1,2,
Jingjing Xia1,2, Wenjun Wu1,2 and Hong Chang1,2 1 National Time Service
Center, Chinese Academy of Sciences (NTSC), Xi’an 710600, People’s Republic of
China  2 School of Astronomy and Space Science, University of Chinese Academy
of Sciences, Beijing 100049, People’s Republic of China<EMAIL_ADDRESS>and<EMAIL_ADDRESS>
###### Abstract

We report the absolute frequency measurement of the $5s^{2}$
${}^{1}S_{0}\rightarrow 5s5p$ ${}^{3}P_{0}$ transition in 87Sr optical lattice
clock (Sr1) at National Time Service Center (NTSC). Its systematic frequency
shifts are evaluated carefully with a total relative uncertainty of $5.1\times
10^{-17}$. The measured absolute frequency is 429 228 004 229 872.91(18) Hz
with a relative uncertainty of $4.13\times 10^{-16}$, with reference to the
ensemble of primary and secondary frequency standards published in the
Circular T bulletin by BIPM through a global navigation satellite system
(GNSS) link.

Keywords: optical clock, SI second, absolute frequency measurement, strontium
lattice clock

  * October 2022

##  1 Introduction

Recent rapid progress of optical clocks based on trapped ions [1-5] and
neutral atoms [6–10] has inspired various applications, such as testing
possible time variation of fundamental constants [11,12], searching the dark
matter [13,14] and applying for relativistic geodesy [15,16]. With estimated
systematic uncertainties of optical clocks now routinely lower than those of
the most accurate Cs fountains [1-10], the Consultative Committee for Time and
Frequency (CCTF) has proposed a roadmap to redefine the second in the
International System of Units (SI) [17]. This roadmap requires the absolute
frequency measurements of three independent optical clocks to independent Cs
primary clocks with fractional uncertainty below $3\times 10^{-16}$. In this
work, we present the systematic evaluation of the 87Sr optical lattice clock
(OLC) and the absolute frequency measurement using a GNSS link to the ensemble
of primary and secondary frequency standards (PSFS) in the BIPM Circular T
bulletin.

This paper is organized as follows. Section 2 outlines the optical lattice
clock apparatus. Section 3 describes the evaluation of systematic frequency
shifts and their uncertainties. Section 4 shows the absolute frequency
measurement of the OLC against International Atomic Time (TAI).

##  2 Experimental setup

Figure 1: (Color online) The clock transition and instability of the 87Sr OLC.
(a) Sideband-resolved spectroscopy. The red line represents the fitting
according to reference [22]. (b) High resolution spectrum with a 0.5 s
spectroscopy pulse. The inset shows the spectrum with a 1.4 s spectroscopy
pulse and the extracted FWHM is 0.59(5) Hz, agreeing with the corresponding
Fourier-limited linewidth of 0.57 Hz. The solid lines indicate Rabi fitting.
(c) Overlapping Allan deviation of the frequency difference between two
interleaved atomic servos at different lattice depths. The comparison data are
divided by $\sqrt{2}$ to reflect the interleaved instability. The red fitting
line (with white frequency noise model) indicates the interleaved instability
of $6.6\times 10^{-16}$ $\rm{(\tau/s)^{-0.5}}$.

The setup of the Sr1 at NTSC has been described elsewhere [18-20]. Here, we
only give a general overview of the setup. 87Sr atoms are evaporated from the
oven and then collimated by a 0.8 mm diameter pinhole and two-dimensional
optical molasses in succession. A Zeeman slower is used to increase the atoms
with a velocity below 50 m/s, which is the maximum velocity for capturing
atoms by the blue magneto-optical trap (MOT). The blue MOT with a magnetic
field gradient of 50 G/cm is operated with light detuned by 40 MHz below the
$5s^{2}$ ${}^{1}S_{0}(F=9/2)\rightarrow 5s5p$ ${}^{1}P_{1}(F=11/2)$ transition
at 461 nm. Limited by the natural linewidth of the 461 nm transition (about
30.2 MHz), the temperature of atoms is typically $\sim$1 mK after the first
stage of cooling by the blue MOT. However, the lattice can capture atoms when
the temperature of atoms is tens of $\mathrm{\mu K}$. Thus, the red MOT,
operated with light detuned by $\sim$500 kHz below the $5s^{2}$
${}^{1}S_{0}(F=9/2)\rightarrow 5s5p$ ${}^{3}P_{1}(F=11/2)$ transition at 689
nm, is used to further reduce atomic temperature to 3 $\mathrm{\mu K}$. To
increase the number of atoms trapped by the red MOT, the light detuned by
$\sim$150 kHz below the $5s^{2}$ ${}^{1}S_{0}(F=9/2)\rightarrow 5s5p$
${}^{3}P_{1}(F=9/2)$ transition at 689 nm is used [18], and the light detuned
by $\sim$50 kHz below this transition is used as the spin-polarized light for
state preparation. At the end of the red MOT, all light beams except the
lattice light are cut off, and about $10^{4}$ atoms are trapped in the
lattice. The lattice is formed by overlapping the incident and it’s retro-
reflected laser beams at the center of the red MOT. The lattice light is
generated by an extended cavity diode laser (ECDL), of which the power is
amplified by a tapered amplifier (TA). Then, the spin-polarized light with
$\sigma^{\pm}$ polarization is used to prepare atoms in $m_{\mathrm{F}}=\pm
9/2$ states. The initial trap depth of the optical lattice is 110
$E_{\rm{R}}$, where $E_{\rm{R}}=h^{2}/2m\lambda^{2}$ is the recoil energy from
a lattice photon, $m$ is the mass of the atom, $h$ is the Planck constant, and
$\lambda\sim 813.42$ nm is the wavelength of the lattice laser. The hotter
atoms are removed by an energy filtering procedure [21], wherein the optical
lattice depth is ramped down within 15 ms to 45 $E_{\rm{R}}$, kept for 20 ms
and then, ramped back to 110 $E_{\rm{R}}$ within 15 ms. After the energy
filtering procedure, approximately 30% of the atoms remain in the lattice.

The ultra-stable clock laser corresponds to the $5s^{2}$
${}^{1}S_{0}\rightarrow 5s5p$ ${}^{3}P_{0}$ transition at 698 nm. Compared to
our previous reported clock instability of $5\times 10^{-15}$
$\rm{(\tau/s)^{-0.5}}$ [18], we have improved the clock instability by about
one order of magnitude by locking the clock laser to a 20 cm ultra-low
expansion (ULE) glass cavity. The vacuum pressure of the ULE cavity is
$2\times 10^{-7}$ Pa maintained by a 100 L/s ion pump. With stabilization of
the power of the cavity transmission light and the temperature of the electro-
optic modulator (EOM), the flicker noise floor of the clock laser is
determined as $4\times 10^{-16}$ by analyzing the in-loop error signal.

The temperature of the ensemble is measured using sideband-resolved
spectroscopy shown in figure 1(a), which reveals an axial temperature of
$T_{\rm{z}}=1.2$ $\rm{\mu K}$ (indicating that 95% atoms populate on the
ground Bloch band) and a radial temperature of $T_{\rm{r}}=2.3$ $\rm{\mu K}$
[22]. Figure 1(b) shows the high-resolution spectrum with a 0.5 s spectroscopy
pulse with a full width at half maximum (FWHM) of 1.6(2) Hz. To explore the
limit of coherence in our clock, we scan the clock transition with a 1.4 s
interrogation time shown in the inset of figure 1(b). Although the FWHM of the
spectrum for 1.4 s clock pulse is 0.59(5) Hz agreeing with the corresponding
Fourier-limited linewidth of 0.57 Hz, the transition probability is about 0.5,
mainly limited by the interaction between atoms and inhomogeneous excitation
due to finite temperature and misalignment angle between the lattice and clock
laser beams [22, 23]. In addition, it is difficult for in-loop operation using
spectrum with linewidth below 1 Hz. Thus, we set the interrogation time to 0.5
s for our regular operation and the corresponding interleaved instability of
$6.6\times 10^{-16}$ $\rm{(\tau/s)^{-0.5}}$ is obtained shown in figure 1(c).

##  3 Systematic shifts of the 87Sr clock transition

There are a number of frequency shifts of the clock transition that must be
evaluated when characterizing a lattice clock’s accuracy. Parts of these
shifts can be measured by the interleaved self-comparison method, such as the
collisional shift, AC Stark shift and DC stark Shift. Other shifts need to
combine the theoretical calculation, such as the blackbody radiation (BBR)
shift and line pulling shift. A summary of the uncertainty contributions is
given in table 1.

Table 1: Corrections and uncertainties of Sr1 during the frequency measurement given in fractional frequencies. All values are in units of $10^{-17}$. Systematic effect | Correction | Uncertainty
---|---|---
BBR chamber | 501.9 | 4.4
BBR oven | 0.72 | 0.1
Lattice scalar/tensor | 5.7 | 2.2
Hyperpolarizability | -1.3 | 0.5
Lattice M1/E2 | 0 | 0.4
Density | 20.32 | 0.84
Tunneling | 0 | 0.6
2nd Zeeman | 9.39 | 0.23
Background gas collisions | 0.51 | 0.11
DC Stark | 1.48 | 0.05
Servo error | 0.037 | 0.046
Line pulling | 0 | 0.002
Probe AC Stark | 0.0093 | 0.0017
Total | 538.77 | 5.1

###  3.1 BBR shift

The BBR shift is due to the differential Stark shift of the clock states from
BBR emitted by the apparatus surrounding the atoms and typically is the
largest frequency correction and systematic uncertainty in state-of-the-art
optical lattice clocks. The BBR shift of a thermal electric field distribution
characterized by a temperature, $T$, can be approximately expressed by [9]

$\displaystyle\Delta\nu\rm{{}_{BBR}}=\nu\rm{{}_{sat}}(\frac{\it{T}}{\it{T}_{0}})^{4}+\nu\rm{{}_{dyn}}[(\frac{\it{T}}{\it{T}_{0}})^{6}+\vartheta(\frac{\it{T}}{T_{0}})^{8}],$
(1)

where $\nu_{\rm{sat}}$= -2.13023(6) Hz [24], $\nu_{\rm{dyn}}$ = -150.51(43)
mHz [25], and $T_{0}$=300 K.

A 10-channel precision temperature scanner (T1001S, XIATECH) and ten
calibrated thin-film platinum resistance thermometers (TFPRTs) are used to
measure the ambient temperature of the atoms. The total uncertainty of the
calibration of TFPRTs is better than 35 mK, where
$0\text{\,}\mathrm{\SIUnitSymbolCelsius}$,
$18\text{\,}\mathrm{\SIUnitSymbolCelsius}$,
$22\text{\,}\mathrm{\SIUnitSymbolCelsius}$,
$24\text{\,}\mathrm{\SIUnitSymbolCelsius}$ and
$30\text{\,}\mathrm{\SIUnitSymbolCelsius}$ are chosen as the calibration
temperatures. Four TFPRTs are put on viewports, covering the top, bottom and
lattice viewport. Five TFPRTs are on the stainless-steel chamber and the last
one monitors room temperature. Once the maximal and minimal temperature
$T_{\rm{max}}$ and $T_{\rm{min}}$ of the system are determined, a
representative temperature and an associated uncertainty can be derived,
assuming a rectangular probability distribution between $T_{\rm{max}}$ and
$T_{\rm{min}}$. During the evaluation, $T_{\rm{max}}$=296.97 K and
$T_{\rm{min}}$=294.81 K are obtained. Thus, we use
($T_{\rm{max}}$+$T_{\rm{min}}$)/2=295.89 K as the ambient temperature and
($T_{\rm{max}}$-$T_{\rm{min}}$)/$\sqrt{12}$ =0.624 K as the measurement
uncertainty [21,26,27]. Therefore, the BBR shift due to the chamber is
evaluated to be $-5.019(44)\times 10^{-15}$.

The BBR shift from the atomic oven is also estimated. The temperature and
solid angle from the oven to MOT are 773(20) K and 0.00009$\rm{\pi}$ sr,
respectively. Thus, the BBR shift from the oven is $-7.2(10)\times 10^{-18}$.
During this evaluation, the heating of the viewport for Zeeman cooling light
is stopped. Thus, we will not consider the extra BBR shift caused by this
viewport.

###  3.2 Density shift

Previous research has verified that in the spin-polarized Fermion ensemble
with a temperature of $\sim$$\rm{\mu K}$, multiply occupied in lattice site
will cause density-dependent shift which is dominated by the $p$ wave
interaction [28]. Density shift relates to the atomic density, excitation
fraction, ensemble temperature and lattice trap depth $U$ [28]. As the
ensemble temperature, trap depth and excitation fraction are constant, the
density shift is in direct proportion to the number of atoms. Density shift
can be measured by the interleaved self-comparison method by modulating the
number of atoms of two interleaved loops [29]. In this work, the number of
atoms is modulated by a factor of 2 through varying the operation time of the
two-dimensional optical molasses between 600 ms and 200 ms, alternatively.
When the excitation fraction is 0.418 and $T_{\rm{z}}=1.2$ $\rm{\mu K}$, the
density shift is measured at different trap depths shown in figure 2. Using
the function of $\alpha_{0}+\alpha U^{1.5}$ to fit the experimental data [6],
the coefficients of $\alpha_{0}$ and $\alpha$ are determined as
$-0.7(40)\times 10^{-6}$ Hz and $-8.94(13)\times 10^{-8}$
Hz/$E_{\rm{R}}^{1.5}$, respectively. Thus, the density shift is
$-2.032(84)\times 10^{-16}$ when the clock regularly runs with $U_{0}$=110
$E_{\rm{R}}$ and $\sim$840 atoms in the lattice.

Figure 2: (Color online) Density shift evaluated at different trap depths
($U$), scaled to a shift with one atom. The solid red line is a fit to the
data with the expected scaling of the shift as $U^{1.5}$. The error bars
indicate 1$\sigma$ uncertainty of the corresponding interleaved measurements.

###  3.3 Lattice AC Stark shift

The lattice laser frequency is locked to a 10 cm long ULE cavity with a
finesse of about 20000 and measured by a commercial fiber optical frequency
comb (OFC, made by IMRA). Before delivering the lattice light to the science
chamber, a volume Bragg grating with a bandwidth of 10 GHz is used to reduce
the shift caused by the TA amplified spontaneous emission (ASE) [30,31]. The
clock laser is co-aligned with the lattice laser. The polarization of the
lattice and clock lasers is cleaned by two linear polarizers (LPVIS100-MP2,
Thorlabs) and carefully aligned with the vertical bias magnetic field by
scanning the $\sigma$ transition spectrum.

During the total measurements, the lattice laser frequency was set to 368 554
485 MHz for suppressing the first-order AC Stark shift [6]. The residual
shift, due to the discrepancy of the tensor shift for different systems [32],
is determined using the interleaved self-comparison method. As the density
shift coefficient relates to the trap depth, we carefully remove the density
shift from the measurement data of the lattice AC Stark shift according to
figure 2. By measuring the frequency difference of the clock transition
between the lattice trap depths of 110 $E_{\rm{R}}$ and 150 $E_{\rm{R}}$, the
scalar/tensor light shift is extrapolated to be $-5.7(22)\times 10^{-17}$ for
our regular trap depth of 110 $E_{\rm{R}}$, where the uncertainty is inferred
from the interleaved instability shown in figure 1(c).

The nonlinear light shift caused by the hyperpolarizability shift and the
magnetic dipole and electric quadrupole shifts (M1/E2) have also been
evaluated. According to the reference [33], the frequency shift coefficients
of hyperpolarizability and M1/E2 terms are 0.46(18) $\rm{\mu
Hz}$/$E_{\rm{R}}^{2}$ and 0(0.31) mHz/$\sqrt{E_{\rm{R}}}$, respectively. In
terms of trap depth of 110 $E_{\rm{R}}$ and a mean vibrational state
occupation of $\overline{n}=0.057$, shifts introduced by the
hyperpolarizability and M1/E2 term are $1.3(5)\times 10^{-17}$ and
$0.0(4)\times 10^{-17}$, respectively.

###  3.4 DC Stark shift

Stray DC electric fields or patch charges on vacuum viewports can induce DC
Stark shift, which was found in optical lattice clocks in 2011 [34]. To remove
possible patch charges on vacuum viewports, we use UV light to illuminate each
viewport. At the beginning of illuminating the window, the vacuum pressure of
the science chamber increases rapidly from $2\times 10^{-9}$ Pa to $1\times
10^{-7}$ Pa. Keeping illuminating for half an hour, the vacuum pressure
returns back to about $5\times 10^{-9}$ Pa. To evaluate the residual DC Stark
shift using the interleaved self-comparison method, we install 3 pairs of
electrodes with a doughnut-like shape to generate the DC electric field. For
each direction, we measure the frequency difference between +100 V and 0 V,
and -100 V and 0 V, respectively. The total residual DC Stark shift is
determined as $-1.48(5)\times 10^{-17}$.

###  3.5 Zeeman shift

The average frequency of $m_{\rm{F}}=-9/2\rightarrow m_{\rm{F}}=-9/2$ and
$m_{\rm{F}}=+9/2\rightarrow m_{\rm{F}}=+9/2$ is used as clock output frequency
of the 87Sr optical lattice clock. To distinguish the two transitions, a large
bias magnetic field is applied. The first-order Zeeman shift is canceled out
by alternatively locking the clock laser between $m_{\rm{F}}=-9/2$ and
$m_{\rm{F}}=+9/2$ components of the clock transition. The frequency gap
between $m_{\rm{F}}=-9/2$ and $m_{\rm{F}}=+9/2$ components of the clock
transition is 405(5) Hz, where the uncertainty indicates the difference
between maximum and minimum frequency gap during 1-month measurements. Thus,
the second order Zeeman shift is $-9.39(23)\times 10^{-17}$ using the reported
coefficient in reference [9].

###  3.6 Other shifts and uncertainties

The site-to-site tunneling of the atoms in the lattice may lead to shift [35].
With the experimental parameters, 99.8% of the atoms populate the two lowest
bands of the lattice (the trap depth is 110 $E_{\rm{R}}$). According to
reference [21,36,37], we use the width of the first excited band of the
lattice as the upper bound for the effect of tunneling, leading to an
uncertainty contribution of $6\times 10^{-18}$.

Collision between background gas and atoms will induce frequency shift in
OLCs. With the measured vacuum lifetime of 5.9(6) s, this shift is estimated
to be $-5.1(11)\times 10^{-18}$ using the reported coefficient in reference
[38].

The clock transition interrogation is interrupted by detuning the clock laser
frequency by 2 MHz from the resonance frequency. This method allows the clock
intensity servo to settle before spectroscopy and eliminates all thermally-
induced differential path length transients in the acousto-optic modulator
(AOM) crystal. The Doppler shift due to path length variation between the
clock laser and the atoms trapped in the lattice is suppressed by locking the
clock laser phase to the lattice reflector. Thus, the shift caused by the AOM
phase chirp and Doppler effect is ignored at the current stage.

We used a second integration of the atomic lock frequency error to compensate
for the frequency drift of the clock laser as proposed in reference [39]. The
servo error is caused by the residual drift of the clock laser. For the whole
measurement time, the mean servo error is $-3.7(46)\times 10^{-19}$ by
averaging the entire history of atomic servo error signal data.

The off-resonant excitation of an impurity spin population can produce a
systematic line pulling frequency shift. By scanning the clock transition
spectrum, the residual total impurity spin population is determined to be less
than 4%. For a 0.5 s $\rm{\pi}$-pulse and a 40.5 Hz splitting between the
$|m_{\rm{F}}|=9/2$ and $|m_{\rm{F}}|=7/2$ clock transitions, the maximum off-
resonant excitation is: $(0.49/0.82)^{2}\times(\pi/0.5)^{2}/((2\pi\texttimes
40.5)^{2}+(\pi/0.5))^{2}$ = $2.176\times 10^{-4}$ [9]. Combining the upper
bound of the atom number in impurity spin states (4%) with this excitation
fraction and the frequency sensitivity of 0.943 Hz [40], an upper bound on the
line pulling effect is $2\times 10^{-20}$. As the laser is locked to the
average frequency of the $\Delta m_{\rm{F}}=0$, which suppresses the shift
caused by the line pulling effect. Thus, we make the value of $2\times
10^{-20}$ be the uncertainty of the line pulling effect.

The probe Stark shift is in direct proportion to the square of $\Omega_{0}$
($\Omega_{0}=2\pi\times 1.3(1)$ Hz for all measurements), where $\Omega_{0}$
indicates the bare states coupling strength between the clock light and atoms.
Thus, the probe Stark shift is $-9.3(17)\times 10^{-20}$ according to the
measured coefficient in our previous publication [41].

###  3.7 Systematic shifts summary

All of the systematic shifts and their uncertainties are summarized in table
1. The total relative systematic shifts correction is $538.77\times 10^{-17}$
and the relative uncertainty is $5.1\times 10^{-17}$.

##  4 Absolute frequency measurement and results

To accurately determine the absolute frequency of the 87Sr OLC at NTSC, the
H-maser (VCH-1003M, No. 5085) is used as the local transfer oscillator. The 10
MHz output of the H-maser is the reference frequency for OFC and all microwave
synthesizers and frequency counters. Both the carrier-envelope offset
frequency ($f_{\rm{ceo}}$=60 MHz) and repetition frequency
($f_{\rm{r}}$$\sim$200 MHz) of the OFC are referenced to the H-maser. The 698
nm output of the 87Sr OLC is delivered to the OFC by a noise-cancelled fiber.
$f_{\rm{r}}$ is divided by a factor of 10 and counted by a phase noise
analyzer (Microchip 53100A). The beat frequency ($f_{\rm{beat}}$) between the
Sr clock output and the corresponding comb tooth is counted by a dual-channel
frequency counter (KEYSIGHT 53230A) shown in figure 3(a). The accuracy of
these frequency counters was assessed by measuring the 10 MHz maser signal,
which is also used as the counter’s external reference, and makes a negligible
contribution to the overall uncertainty.

Figure 3: (Color online) Optical clock frequency measured with the OFC and the
instability of the H-maser. (a) shows the beat frequency $f_{\rm{beat}}$
between the frequency of the 87Sr OLC transition and the OFC as measured over
30 days. (b) The fractional frequency instability of $f_{\rm{beat}}$ as
measured by the overlapping Allan deviation. The Allan deviation points are
based on datasets over MJD 59801 to 59810 (with an up-time ratio of 89%). From
this instability data, we extract the stochastic noise model of the H-maser
indicated by the red curve.

The H-maser is continually compared to the UTC(NTSC), which is the real-time
realization of the coordinated universal time (UTC) by the NTSC, with a local
time difference measurement system. UTC(NTSC) has a GNSS link to TAI.
According to the monthly Circular T bulletin, the differences between
UTC(NTSC), TAI and the best realization of the SI second by PSFS can be
determined. Considering inevitable dead times in the measurements, the
procedure linking the local optical frequency reference to the SI second can
be summarized by the following expression [42,43]

$\displaystyle\frac{\rm{y_{Sr}}}{\rm{y_{SIs}}}=\frac{\rm{y_{Sr}}(\it{\Delta
t}\rm{{}_{1})}}{\rm{y_{HM}}(\it{\Delta
t}\rm{{}_{1})}}\times\frac{\rm{y_{HM}}(\it{\Delta
t}\rm{{}_{1})}}{\rm{y_{HM}}(\it{\Delta
t}\rm{{}_{2})}}\times\frac{\rm{y_{HM}}(\it{\Delta
t}\rm{{}_{2})}}{\rm{y_{UTC(NTSC)}}(\it{\Delta t}\rm{{}_{2})}}$
$\displaystyle\quad\quad\quad\times\frac{\rm{y_{UTC(NTSC)}}(\it{\Delta
t}\rm{{}_{2})}}{\rm{y_{TAI}}(\it{\Delta
t}\rm{{}_{2})}}\times\frac{\rm{y_{TAI}}(\it{\Delta
t}\rm{{}_{2})}}{\rm{y_{TAI}}(\it{\Delta
t}\rm{{}_{3})}}\times\frac{\rm{y_{TAI}}(\it{\Delta
t}\rm{{}_{3})}}{\rm{y_{SIs}}(\it{\Delta t}\rm{{}_{3})}},$ (2)

where $\rm{y_{Sr}}$, $\rm{y_{HM}}$, $\rm{y_{UTC(NTSC)}}$, $\rm{y_{TAI}}$ and
$\rm{y_{SIs}}$ represent the rates of the 87Sr OLC, H-maser, UTC(NTSC), TAI
and the SI second (1 Hz), respectively. $\it{\Delta t}\rm{{}_{i}}$ (i = 1, 2,
3) denotes the effective measurement time intervals of the ratios. $\it{\Delta
t}\rm{{}_{1}}$ is the sum of all effective operation time of Sr1 and OFC
measurements; $\it{\Delta t}\rm{{}_{2}}$ indicates the minimum range covering
$\it{\Delta t}\rm{{}_{1}}$, of which the endpoints satisfy the 5-days interval
reported in Circular T; $\it{\Delta t}\rm{{}_{3}}$ represents the 1-month TAI
reporting period.

Table 2: The corrections and uncertainties of the calibration of the H-maser frequency. All values are in units of $10^{-16}$. Ratio | Correction | Uncertainty
---|---|---
$\rm{y_{HM}}(\it{\Delta t}\rm{{}_{1})}/\rm{y_{HM}}(\it{\Delta t}\rm{{}_{2})}$ | -0.083 | 3.1
$\rm{y_{HM}}(\it{\Delta t}\rm{{}_{2})}/\rm{y_{UTC(NTSC)}}(\it{\Delta t}\rm{{}_{2})}$ | 1258.8 | 0.1
$\rm{y_{UTC(NTSC)}}(\it{\Delta t}\rm{{}_{2})}/\rm{y_{TAI}}(\it{\Delta t}\rm{{}_{2})}$ | 1.54 | 1.96
$\rm{y_{TAI}}(\it{\Delta t}\rm{{}_{2})}/\rm{y_{TAI}}(\it{\Delta t}\rm{{}_{3})}$ | - | -
$\rm{y_{TAI}}(\it{\Delta t}\rm{{}_{3})}/\rm{y_{SIs}}(\it{\Delta t}\rm{{}_{3})}$ | -0.9 | 1.2
Total | 1259.36 | 3.9

The instability of the comparison between the H-maser and Sr1 is shown in
figure 3(b), which can be used to infer the noise model of the H-maser as the
instability of Sr1 is smaller than the H-maser by more than two orders of
magnitude [25,26,43]. The linear drift rate of the H-maser, considered
constant during the one-month measurement, is extracted as $-6.4\times
10^{-17}$ $\rm{d^{-1}}$ from the optical frequency measurements. After
removing the linear frequency drift, we extracted the four noise contributions
of the H-maser with a white phase noise part of $2.21\times 10^{-13}$
$(\tau/s)^{-1}$, a white frequency noise part of $3.05\times 10^{-13}$
$(\tau/s)^{-0.5}$, a flicker frequency noise part of $6.01\times 10^{-16}$
$(\tau/s)^{0}$ and a random walk frequency noise part of $4.49\times 10^{-19}$
$(\tau/s)^{0.5}$. Based on this H-maser noise model, we simulated one hundred
time series over measurements period $\it{\Delta t}\rm{{}_{1}}$ to $\it{\Delta
t}\rm{{}_{2}}$ using the software of Stable32 [44, 45], and the extrapolation
uncertainty of the H-maser frequency is evaluated to be $3.1\times 10^{-16}$.
The optical frequency measurements are made between MJD 59789.13–59819, which
is $\sim$0.13 days shorter than the period covered by $\it{\Delta
t}\rm{{}_{2}}$. The linear drift of H-maser gives a correction of $-8.3\times
10^{-18}$. Combining the dead time and drift of the H-maser, the correction of
$\rm{y_{HM}}(\it{\Delta t}\rm{{}_{1})}/\rm{y_{HM}}(\it{\Delta t}\rm{{}_{2})}$
is $-8.3\times 10^{-18}$, with an uncertainty of $3.1\times 10^{-16}$. The
correction of $\rm{y_{HM}}(\it{\Delta
t}\rm{{}_{2})}/\rm{y_{UTC(NTSC)}}(\it{\Delta t}\rm{{}_{2})}$ is calculated to
be $1.2588(1)\times 10^{-13}$ from local DMTD measurements. As $\it{\Delta
t}\rm{{}_{3}}=\it{\Delta t}\rm{{}_{2}}$, it is unnecessary to evaluate the
correction of $\rm{y_{TAI}}(\it{\Delta t}\rm{{}_{2})}/\rm{y_{TAI}}(\it{\Delta
t}\rm{{}_{3})}$. With the data from Circular T 416, the corrections of
$\rm{y_{UTC(NTSC)}}(\it{\Delta t}\rm{{}_{2})}/\rm{y_{TAI}}(\it{\Delta
t}\rm{{}_{2})}$ is calculated to be $1.54\times 10^{-16}$, of which the
uncertainty can be determined by $(5/30)^{0.9}\times\sqrt{2}\times
0.3/86400/5$ = $1.96\times 10^{-16}$ [26]. The correction of
$\rm{y_{TAI}}(\it{\Delta t}\rm{{}_{3})}/\rm{y_{SIs}}(\it{\Delta
t}\rm{{}_{3})}$ is $-0.9(12)\times 10^{-16}$. The calibration of the H-maser
has a correction factor of $1.25936\times 10^{-13}$ with an uncertainty of
$3.9\times 10^{-16}$, as summarized in table 2.

Table 3: The corrections and uncertainties of the absolute frequency measurement. All values are in units of $10^{-16}$. Ratio | Correction | Uncertainty
---|---|---
Sr1 systematic | 53.877 | 0.51
Statistical | 0 | 1.1
Relativistic redshift | -523.7 | 0.6
H-maser calibration | 1263.03 | 3.9
Total | 793.207 | 4.13

The standard error of the weighted mean of all measurements shown in figure
3(a) is $1.1\times 10^{-16}$, which has been scaled by the root of the reduced
chi-square $\chi_{red}^{2}$ ($\chi_{red}^{2}=2.8$) and used to be the
statistical uncertainty of the optical frequency measurement. The orthometric
height of the lattice is 480.3(5) m by tracing to a leveling benchmark inside
the building on our campus. This leveling benchmark is referenced to the China
1985 national height datum [46]. Benefiting from the one-month-long
measurement with an up-time ratio of 57.1%, the time-variable gravity
potential component is averaged out and considerable to be negligible [47].
With the method in reference [48], the fractional relativistic redshift is
$523.7(6)\times 10^{-16}$. Thus, the total calibration of the whole
measurement system is $793.207\times 10^{-16}$ with an uncertainty of
$4.13\times 10^{-16}$. The uncertainties of the absolute frequency measurement
are summarized in table 3. The absolute frequency of Sr1 is 429 228 004 229
872.91(18) Hz, which is consistent with the 2021 CIPM recommended value of the
87Sr neutral atom optical frequency of 429 228 004 229 872.99(7) Hz.

##  5 Conclusion

In summary, we have realized an 87Sr optical lattice clock with an estimated
systematic uncertainty of $5.1\times 10^{-17}$ and determined its absolute
frequency to be 429 228 004 229 872.91(18) Hz by tracing its frequency to the
fountain ensemble in Circular T through a GNSS link with an overall
uncertainty of $4.13\times 10^{-16}$. Future works will focus on improving the
effective operation time of the optical lattice clock, reducing the systematic
uncertainty and comparing against other optical clocks with a comparison
uncertainty below the $1\times 10^{-17}$ level.

##  6 Acknowledgments

This work is supported by the Strategic Priority Research Program of the
Chinese Academy of Sciences (Grant No. XDB35010202) and the National Natural
Science Foundation of China (No. 12203057).

##  7 Data availability

The data that support the findings of this study are available upon reasonable
request from the authors.

##  8 References

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