# Global Field Totients

Santiago Arango-Piñeros  Department of Mathematics, Emory University, Atlanta,
GA 30322<EMAIL_ADDRESS>and  Juan Diego Rojas  Departmento de
Mathematicas, Universidad de Los Andes, Bogotá, Colombia
<EMAIL_ADDRESS>
###### Abstract.

Using a remainder theorem for valuations of a field, we give a new perspective
on the norm function of a global field. We define the Euler totient function
of a global field and recover the essential analytical properties of the
classical arithmetical function, namely the product formula. In addition, we
prove the holomorphicity of the associated zeta function. As an application,
we recover the analog of the mean value theorem of Erdős, Dressler, and
Bateman via the Weiner-Ikehara theorem.

##  1\. Introduction

###  1.1. Euler’s function

The classical arithmetical function of Euler, denoted by $\varphi$, is defined
in group theoretic terms as the size of the group of units modulo $n$. The
Chinese remainder theorem implies that $\varphi(n)$ is a multiplicative
arithmetical function. This means that for every pair of relatively prime
positive integers $a$ and $b$, $\varphi(ab)=\varphi(a)\varphi(b)$. Since for
every prime number $p$ and every positive integer $r$ the ring
$\operatorname{\mathbb{Z}}/p^{r}\operatorname{\mathbb{Z}}$ is local with
maximal ideal $p\operatorname{\mathbb{Z}}/p^{r}\operatorname{\mathbb{Z}}$, the
units modulo $p^{r}$ correspond to the complement of
$p\operatorname{\mathbb{Z}}/p^{r}\operatorname{\mathbb{Z}}$ inside
$\operatorname{\mathbb{Z}}/p^{r}\operatorname{\mathbb{Z}}$. This implies
$\varphi(p^{r})=p^{r}(1-1/p)$. The unique factorization property of the
integers yields the product formula

$\displaystyle\varphi(n)=n\prod_{p\mid n}\left(1-\frac{1}{p}\right)\,.$

The product formula is the soul of several results in classical analytic
number theory concerning the Euler function. For instance, the mean value
theorem of Erdős, Dressler, and Bateman, which we proceed to recall.

A positive integer $n$ is called a totient if belongs to the image of
$\varphi$. The totient multiplicity of an integer $n$, denoted by $t(n)$, is
the size of the fiber $\varphi^{-1}(n)$. Since $\varphi(n)$ goes to infinity
with $n$, $t(n)$ is always finite. The study of the distribution of totient
numbers is an alluring research problem studied by many. In particular, the
calculation of the mean value of the arithmetical function $t(n)$ was first
approached by Erdős in [Erd45]. Using techniques from probabilistic number
theory, he showed the existence of the limit

$\displaystyle\lim_{N\to\infty}\,\dfrac{1}{N}\sum_{n=1}^{N}t(n)=\alpha\,.$

Twenty five years later, Dressler [Dre70] explicitly calculated the limit,
obtaining the remarkable constant

$\displaystyle\alpha=\zeta(2)\zeta(3)/\zeta(6)\,,$

where $\zeta(s)$ is, of course, the Riemann zeta function. Bateman [Bat72]
gave a second proof of this via the Wiener-Ikehara theorem. We summarize this
joint effort in the following theorem.

######  Theorem 1 (Erdős, Dressler, Bateman).

Let $t(n)$ denote the totient multiplicity arithmetical function, and let
$\zeta(s)$ denote the Riemann zeta function. Then, the mean value of $t(n)$ is
given by

$\displaystyle\lim_{N\to\infty}\dfrac{1}{N}\sum_{n=1}^{N}t(n)=\zeta(2)\zeta(3)/\zeta(6)\,.$

###  1.2. Results

The core of this manuscript is a theorem for discrete valuation rings which is
reminiscent of the Chinese remainder theorem. The number theoretic results
will follow smoothly from this algebraic fact. We refer the reader to Section
2 for the complete statement. We where unable to find this result in the
literature, so we provide a proof here.

######  Theorem A (DVR remainder theorem).

Let $L$ be a field. If $V$ and $W$ are finite disjoint multisets of discrete
valuations of $L$, then

$\displaystyle(R_{V}\cap R_{W})/(I_{V}\cap
I_{W})\cong(R_{V}/I_{V})\times(R_{W}/I_{W})\,.$

The main result of this note is the correct definition of the Euler function
$\varphi_{K}$ attached to a global field $K$. We claim it is the correct one
because the expected analytic properties follow comfortably from it. In
particular, we recover the product formula. We refer the reader to Subsection
1.3 to consult the unrevealed notation.

######  Theorem B (Product formula).

Let $K$ be a global field with Euler function $\varphi_{K}$ and norm
$\mathsf{N}_{K}$. Then, for every nonzero effective divisor
$D\in\operatorname{\mathsf{Div}}^{+}(X)$,

$\displaystyle\varphi_{K}(D)=\mathsf{N}_{K}(D)\prod_{P\leq
D}(1-\mathsf{N}_{K}(P)^{-1})\,.$

Following Bateman, we prove the holomorphicity of the totient zeta function
$T_{K}(s)$ attached to $K$. Furthermore, we show that this function is
intimately related to the Weil zeta function $\zeta_{K}(s)$ of the field.

######  Theorem C.

Let $K$ be a global field. The totient zeta function $T_{K}(s)$ attached to
$K$ converges absolutely and uniformly in the open half plane $\sigma>1$.
Furthermore, one has the identity

$\displaystyle\dfrac{T_{K}(s)}{\zeta_{K}(s)}=f_{K}(s)\colonequals\prod_{P}\left[1+(\mathsf{N}_{K}(P)-1)^{-s}-\mathsf{N}_{K}(P)^{-s}\right]\,,$

where the product ranges over all closed points $P\in X$ and $f_{K}(s)$ is
holomorphic on the open half plane $\sigma>0$.

To conclude, we apply the Weiner-Ikehara theorem to the totient zeta function
of $K$ and generalize Theorem 1 to the global field case.

######  Theorem D (Mean value of global field totients).

Let $K$ be a global field, and let $t_{K}(n)$ denote the totient multiplicity
arithmetical function of $K$. Denote by $\rho_{K}$ the residue of the Weil
zeta function $\zeta_{K}(s)$ at $s=1$. Then, the mean value of $t_{K}(n)$

$\displaystyle\lim_{N\to\infty}\dfrac{1}{N}\,\sum_{n=1}^{N}t_{K}(n)\,=\,\dfrac{\rho_{K}\,\zeta_{K}(2)\zeta_{K}(3)}{\zeta_{K}(6)}\,.$

###  1.3. Notation

First of all, for a perfect field $k$, a $k$-variety is a separated scheme of
finite type over $k$. We use the letter $q$ to denote a fixed prime power, and
the ground field of the varieties considered is always a finite field with $q$
elements $\operatorname{\mathbb{F}}_{q}$. The letter $K$ denotes a global
field, and $X$ denotes the associated scheme. By this we mean the following:

  * •

If $K$ is a number field, $X$ is the prime spectrum of the ring of integers
${\mathcal{O}}_{K}$.

  * •

If $K$ is a global function field over $\operatorname{\mathbb{F}}_{q}$, $X$ is
the smooth projective model corresponding to $X$ via resolution of
singularities. Recall that $X$ is a smooth integral projective
$\operatorname{\mathbb{F}}_{q}$-variety of dimension $1$.

$\operatorname{\mathsf{Div}}(X)$ is the group of Weil divisors of $X$ and we
denote by $\operatorname{\mathsf{Div}}^{+}(X)$ the semigroup of effective
divisors. We restrict the use of $P$ for irreducible divisors. For every
irreducible divisor $P$,
$\nu_{P}\colon\operatorname{\mathsf{Div}}(X)\to\operatorname{\mathbb{Z}}$ is
the order function, which extends the discrete valuation of the stalk
${\mathcal{O}}_{X,P}$. The support of a divisor
$D\in\operatorname{\mathsf{Div}}(X)$, denoted by
$\operatorname{\mathsf{Supp}}D$, is the finite set of irreducible divisors $P$
such that $\nu_{P}(D)\neq 0$. We abbreviate the statement
$P\in\operatorname{\mathsf{Supp}}D$ by $P\leq D$.

######  Definition 2 (Arithmetic functions on divisors).

  1. (1)

Two effective divisors $D_{1},D_{2}$ are relatively prime if
$(\operatorname{\mathsf{Supp}}D_{1})\cap(\operatorname{\mathsf{Supp}}D_{2})=\emptyset$.

  2. (2)

A function
$f\colon\operatorname{\mathsf{Div}}^{+}(X)\to\operatorname{\mathbb{Z}}$ is
completely multiplicative if $f(D_{1}+D_{2})=f(D_{1})f(D_{2})$ for every pair
of divisors $D_{1},D_{2}$.

  3. (3)

A function
$f\colon\operatorname{\mathsf{Div}}^{+}(X)\to\operatorname{\mathbb{Z}}$ is
multiplicative if $f(D_{1}+D_{2})=f(D_{1})f(D_{2})$ for every pair of
relatively prime divisors $D_{1},D_{2}$.

We use the letter $s$ to denote a complex variable. As is customary in
analytic number theory, $\sigma$ and $t$ denote the real and imaginary part of
$s$ respectively. Finally, we denote by $\zeta_{K}(s)$ the Weil zeta function
of $X$.

##  2\. A remainder theorem for discrete valuations

The motivation for this section comes from the aspiration of unifying the
definition of the norm (and Euler function) of a global field. This will
become clear after Section 3, although we consider the result to be
interesting on its own. We follow the definitions and conventions in Altman
and Kleiman’s book [AK13, Chapter 23]. In particular, all discrete valuations
are surjective.

In this section $L$ will denote a field. For a discrete valuation $v$ of $L$,
$R_{v}\colonequals\\{x\in L\colon v(x)\geq 0\\}$ denotes the corresponding
discrete valuation ring, and $\mathfrak{m}_{v}\colonequals\\{x\in L\colon
v(x)>0\\}$ denotes its maximal ideal. For every $v$, we fix a uniformizer
$t_{v}$ at $v$.

A multiset is a pair $(A,\mu)$ where $A$ is a set and $\mu\colon
A\to\operatorname{\mathbb{Z}}^{+}$ is a function, called the multiplicity
function of $A$. We omit the multiplicity function from the notation and use
$\mu$ to denote the multiplicity function of all multisets. We consider
multisets of discrete valuations of the field $L$ and attach certain subrings
and ideals to them.

######  Definition 3.

Let $L$ be a field and let $V$ be a multiset of discrete valuations of $L$.

  1. (1)

Define the ring associated to $V$ by

$\displaystyle R_{V}\colonequals\bigcap_{v\in V}R_{v}\,.$

  2. (2)

Define the ideal associated to $V$ by

$\displaystyle I_{V}\colonequals\bigcap_{v\in V}\mathfrak{m}_{v}^{\mu(v)}\,.$

See A

###### Proof.

The composition of the inclusion with the natural projection gives morphisms
from $R_{V}\cap R_{W}$ to the quotients $R_{V}/I_{V}$ and $R_{W}/I_{W}$.
Therefore, the map that takes an element in the intersection $f\in R_{V}\cap
R_{W}$ and assigns the ordered pair $(f+I_{V},f+I_{W})$ is a ring
homomorphism. Furthermore, its kernel is precisely $I_{V}\cap I_{W}$. In order
to complete the proof, we need to show that this homomorphism is surjective.
Take an element $(f_{1}+I_{V},f_{2}+I_{W})$, and define the positive integers

$\displaystyle N\colonequals\max_{v\in V}\\{\mu(v),-v(f_{2})\\}\,\quad
M\colonequals\max_{w\in W}\\{\mu(w),-w(f_{1})\\}\,.$

Define $t_{V}\colonequals\prod_{v\in V}t_{v}$ and
$t_{W}\colonequals\prod_{w\in W}t_{w}$ and let $f\colonequals
f_{1}t_{W}^{M}+f_{2}t_{V}^{N}\in L$. Then for every $v\in V$ and $w\in W$ we
have the inequalities

$\displaystyle\nu(f)\geq\min\\{\nu(f_{1}),\nu(f_{2})+N\\}\geq 0\,,\quad
w(f)\geq\min\\{w(f_{1})+M,w(f_{2})\\}\geq 0\,.$

This implies that $f\in R_{\nu}\cap R_{w}$. By construction,
$f+I_{V}=f_{1}+I_{V}$ and $f+I_{W}=f_{2}+I_{W}$. ∎

##  3\. The global field Euler function

The norm of a closed point in $X$ is the dimension of the residue field
$k(P)$. One then defines the norm of an effective divisor by extending
multiplicatively. When $K$ is a number field and $I$ is an integral ideal, the
norm of $I$ coincides with the cardinality of the finite ring
${\mathcal{O}}_{K}/I$. When $K$ is a global function field over
$\operatorname{\mathbb{F}}_{q}$, the norm of an effective divisor $D$ is
$q^{\deg D}$. In this section we introduce a new perspective that unifies the
proofs of our results.

Note that every divisor $D\in\operatorname{\mathsf{Div}}(X)$ corresponds to a
finite multiset of closed points in $X$, where the multiplicity of each closed
point $P$ is $\nu_{P}(D)$. Since $X$ is smooth, each closed point corresponds
to a discrete valuation of $K$. Following Definition 3, for an effective
divisor $D\in\operatorname{\mathsf{Div}}^{+}(X)$, we denote by
${\mathcal{O}}_{D}$ the ring associated to the multiset***This approach was
suggested to us by Guillermo Mantilla-Soler. of valuations given by $D$.
Similarly, $I_{D}$ denotes the ideal associated to the multiset of valuations
given by $D$. Explicitly,

$\displaystyle{\mathcal{O}}_{D}\colonequals\bigcap_{P\in\,\operatorname{\mathsf{Supp}}D}{\mathcal{O}}_{X,P}\,,\quad
I_{D}\colonequals\bigcap_{P\in\,\operatorname{\mathsf{Supp}}D}\mathfrak{m}_{P}^{\nu_{P}(D)}\,.$

######  Definition 4 (Numerical norm).

Let $K$ be a global field. The norm of $K$ is the function
$\mathsf{N}_{K}\colon\operatorname{\mathsf{Div}}^{+}(X)\to\operatorname{\mathbb{Z}}$
given by

$\mathsf{N}_{K}(D)\colonequals\begin{cases}1,&\text{ for }D=0,\\\
\\#({\mathcal{O}}_{D}/I_{D}),&\text{ otherwise.}\end{cases}$

Applying Theorem A to a global field $K$, we show that the perspective
introduced in Definition 4 coincides with the usual definition.

######  Corollary 5 (Properties of the norm).

Let $K$ be a global field. Then, for every nonzero effective divisor
$D\in\operatorname{\mathsf{Div}}^{+}(X)$, we have the ring isomorphism

$\displaystyle{\mathcal{O}}_{D}/I_{D}\cong\prod_{P\leq
D}{\mathcal{O}}_{X,P}/\mathfrak{m}_{P}^{\nu_{P}(D)}\,.$

Moreover,

  1. a)

When $K$ is a number field, and $I$ is the ideal $I\colonequals\prod_{P\leq
D}P^{\nu_{P}(D)}$ the rings ${\mathcal{O}}_{D}/I_{D}$ and
${\mathcal{O}}_{K}/I$ are isomorphic. In particular,
$\mathsf{N}_{K}(D)=\\#({\mathcal{O}}_{K}/I)$.

  2. b)

When $K$ is a global function field over $\operatorname{\mathbb{F}}_{q}$,
$\mathsf{N}_{K}(D)=q^{\deg D}$.

In particular,
$\mathsf{N}_{K}\colon\operatorname{\mathsf{Div}}^{+}(X)\to\operatorname{\mathbb{Z}}$
is a completely multiplicative function on divisors.

###### Proof.

The isomorphism follows from Theorem A to this context. Item (a) follows from
the isomorphism

$\displaystyle{\mathcal{O}}_{D}/I_{P}^{n}={\mathcal{O}}_{D}/\mathfrak{m}_{P}^{n}\cong({\mathcal{O}}_{K})_{P}/P^{n}({\mathcal{O}}_{K})_{P}\cong{\mathcal{O}}_{K}/P^{n}\,,$

for every maximal ideal $P\lhd{\mathcal{O}}_{K}$, and the Chinese remainder
theorem. Item (b) follows from the observation that for a closed point $P\in
X$, ${\mathcal{O}}_{D}={\mathcal{O}}_{X,P}$ and $I_{P}=\mathfrak{m}_{P}$, so
that ${\mathcal{O}}_{P}/I_{P}$ is precisely the residue field of $P$. ∎

The author’s first guess for the definition of the Euler function of a global
field was the same from elementary number theory. In other words, we assumed
that $\varphi_{K}(D)$ should count the number of effective divisors with norm
not exceeding $\mathsf{N}_{K}(D)$, which are coprime to $D$. Even though this
approach works in the number field case, the reader may check that this naive
Euler function is not even multiplicative in the case of ${\mathbb{P}}^{1}$.
Inspired by the group theoretic definition from Section 1, Definition 4, and
Corollary 5, we propose the following definition.

######  Definition 6 (Global field Euler function).

Let $K$ be a global field. The Euler function of $K$ is the function
$\varphi_{K}\colon\operatorname{\mathsf{Div}}^{+}(X)\to\operatorname{\mathbb{Z}}$
given by

$\varphi_{K}(D)\colonequals\begin{cases}1,&\text{ for }D=0,\\\
\\#({\mathcal{O}}_{D}/I_{D})^{\times},&\text{ otherwise.}\end{cases}$

In particular, for every closed point $P\in X$,
$\varphi_{K}(P)=\mathsf{N}_{K}(P)-1$. By Corollary 5, $\varphi_{K}$ is
multiplicative. The product formula will follow from the calculation of the
units in the ring ${\mathcal{O}}_{rP}/I_{rP}$ for $P\in X$ closed and $r\geq
1$.

See B

###### Proof.

Take a closed point $P\in X$, and let $r\geq 1$. Consider the divisor $D=rP$,
and note that ${\mathcal{O}}_{D}={\mathcal{O}}_{X,P}$ and
$I_{D}=\mathfrak{m}_{P}^{r}$. The ring
${\mathcal{O}}_{D}/I_{D}={\mathcal{O}}_{X,P}/\mathfrak{m}_{P}^{r}$ is local
with maximal ideal $\mathfrak{m}_{P}/\mathfrak{m}_{P}^{r}$. Therefore,
$({\mathcal{O}}_{X,P}/\mathfrak{m}_{P}^{r})^{\times}={\mathcal{O}}_{X,P}/\mathfrak{m}_{P}^{r}-\mathfrak{m}_{P}/\mathfrak{m}_{P}^{r}$.
From the exact sequence

$\displaystyle
0\longrightarrow\mathfrak{m}_{P}/\mathfrak{m}_{P}^{r}\longrightarrow{\mathcal{O}}_{X,P}/\mathfrak{m}_{P}^{r}\longrightarrow{\mathcal{O}}_{X,P}/\mathfrak{m}_{P}\longrightarrow
0\,,$

we deduce that
$\\#(\mathfrak{m}_{P}/\mathfrak{m}_{P}^{r})=\mathsf{N}_{K}(P)^{r-1}$, and

$\varphi_{K}(rP)=\mathsf{N}_{K}(P)^{r}-\mathsf{N}_{K}(P)^{r-1}=\mathsf{N}_{K}(rP)(1-\mathsf{N}_{K}(P)^{-1}).$

The result follows for general divisors from this calculation and Corollary 5.
∎

##  4\. Holomorphicity of the totient zeta function

Just as the Weil zeta function of $K$ is the series associated to the norm
$\mathsf{N}_{K}$

$\displaystyle\zeta_{K}(s)=\sum_{D\in\operatorname{\mathsf{Div}}^{+}(X)}\mathsf{N}_{K}(D)^{-s}\,,\quad\text{
when }\sigma>1\,,$

it is natural to consider the series associated to the Euler function
$\varphi_{K}$. Following Bateman, we define the totient zeta function as the
formal series

$\displaystyle
T_{K}(s)\colonequals\sum_{D\in\operatorname{\mathsf{Div}}^{+}(X)}\varphi_{K}(D)^{-s}\,.$

In this section we show that $T_{K}(s)$ converges absolutely and uniformly on
$\sigma>1$. This will follow from the convergence of $\zeta_{K}(s)$ and the
convergence of an Euler product that appears spontaneously as a factor of
$T_{K}(s)$.

######  Lemma 7.

Let $K$ be a global field. For each closed point $P\in X$, consider the
function $F_{P}(s)\colonequals
1+(\mathsf{N}_{K}(P)-1)^{-s}-\mathsf{N}_{K}(P)^{-s}$. The Euler product
$\prod_{P}F_{P}(s)$ converges uniformly to an holomorphic function $f_{K}(s)$
on the open half plane $\sigma>0$.

###### Proof.

We drop the subindex $K$ in the norm and Euler function for the sake of
clarity. Fix $\delta>0$ and let $U_{\delta}$ be the open half plane
$\sigma>\delta$. We have the trivial lower bound $\mathsf{N}(D)\geq 2$ for
every divisor nonzero effective divisor $D$. By choosing the principal branch
of the logarithm, $F_{P}(s)$ is holomorphic in $U_{\delta}$. Thus
$\left(F_{P}(s)\right)_{P}$ is a sequence of holomorphic functions on
$U_{\delta}$. Fix $P$ and denote momentarily $x=\mathsf{N}(P)$. Then

(1)
$|F_{P}(s)-1|=|(x-1)^{-s}-x^{-s}|=\Big{|}s\int_{x-1}^{x}z^{-s-1}\,\mathrm{d}z\Big{|}\leq|s|(x-1)^{-\sigma-1}\,.$

By Inequality 1, we have that for every $P$,

$\displaystyle|F_{P}(s)-1|\ll\mathsf{N}(P)^{-\delta-1}\,,$

uniformly on every compact subset of $U_{\delta}$. Since

$\displaystyle\sum_{P}|F_{P}(s)-1|\ll\sum_{P}\mathsf{N}(P)^{-\delta-1}\leq\zeta_{K}(\delta+1)<\infty\,,$

we conclude that the infinite product $\prod_{P}F_{P}$ converges locally
uniformly on $U_{\delta}$, for every $\delta>0$, and therefore represents an
holomorphic function $f_{K}(s)$ on the half plane $\sigma>0$. ∎

The multiplicativity of $\varphi_{K}$ allows us to manipulate the formal
series $T_{K}(s)$ to get a familiar function, of which we know the legitimacy
of its convergence. From this informal idea we get the following theorem.

See C

###### Proof.

We drop the subindex $K$ in the norm and Euler function for the sake of
clarity. Consider the following manipulation of formal series and products.

$\displaystyle T_{K}(s)$ $\displaystyle\colonequals\,\sum_{D\geq
0}\varphi(D)^{-s}$ (By the multiplicativity of $\varphi$)
$\displaystyle=\,\prod_{P}\sum_{m=0}^{\infty}\varphi(mP)^{-s}$ (By Theorem B)
$\displaystyle=\,\prod_{P}\Big{(}1+\sum_{m=1}^{\infty}\Big{[}\mathsf{N}(P)^{m-1}(\mathsf{N}(P)-1)\Big{]}^{-s}\Big{)}.$
Observe that the product converges for $\sigma>1$, and therefore we have
absolute convergence of $T_{K}(s)$ in this half plane. (Factor
$(\mathsf{N}(P)-1)^{-s}$ and put $n=m-1$)
$\displaystyle=\,\prod_{P}\Big{(}1+(\mathsf{N}(P)-1)^{-s}\sum_{n=0}^{\infty}\mathsf{N}(P)^{-ns}\Big{)}$
(Geometric series)
$\displaystyle=\,\prod_{P}\Big{(}1+(\mathsf{N}(P)-1)^{-s}(1-\mathsf{N}(P)^{-s})^{-1}\Big{)}$
(Factoring the zeta function)
$\displaystyle=\zeta_{K}(s)\prod_{P}\Big{(}1+(\mathsf{N}(P)-1)^{-s}-\mathsf{N}(P)^{-s}\Big{)}$
$\displaystyle=\zeta_{K}(s)f_{K}(s)\,.$

By Lemma 7, we have that uniform convergence holds in $\sigma>1$. Therefore,
the rearrangements made are all justified in this domain, yielding the desired
result. In particular, rewriting $T_{K}(s)$ as a Dirichlet series we get that
the totient multiplicity arithmetical function $t_{K}(n)$ is well defined. ∎

##  5\. Mean value of global field totients

In this section we replicate Bateman’s argument to calculate the mean value of
the totient multiplicity arithmetical function $t_{\mathsf{K}}(n)$ attached to
the global field $K$. The proof of this result relies heavily on the
holomorphicity of $T_{K}(s)$ and the Wiener-Ikehara theorem. The original
references are [Ike31] and [Wie32]. Also, the book [MV07, Section 8.3]
contains a complete discussion of this result. We recall the statement here
for convenience.

######  Theorem 8 (Wiener-Ikehara).

Let $(a_{n})_{n=1}^{\infty}$ be a sequence of non negative real numbers such
that the associated Dirichlet series $\sum a_{n}n^{-s}$ converges in the open
half plane $\sigma>1$. If there exists a constant $\alpha$ and a continuous
function $h(s)$ on the closed half plane $\sigma\geq 1$ such that

$h(s)=\left(\sum_{n=1}^{\infty}\frac{a_{n}}{n^{s}}\right)-\left(\frac{\alpha}{s-1}\right)\,$

on $\sigma>1$, then

$\lim_{N\to\infty}\,\frac{1}{N}\,\sum_{n=1}^{N}a_{n}\,=\,\alpha\,.$

######  Observation 1.

In particular, the hypothesis of the theorem are satisfied when $\sum
a_{n}n^{-s}$ is meromorphic in $\sigma\geq 1$ with a unique (simple) pole at
$s=1$ of residue $\alpha$.

See D

###### Proof.

By Theorem C, we have the equality

$\displaystyle\sum_{m=1}^{\infty}\frac{t_{\mathsf{K}}(m)}{m^{s}}\,=\,\zeta_{\mathsf{K}}(s)f_{\mathsf{K}}(s)\,.$

A direct calculation gives
$f_{\mathsf{K}}(1)=\zeta_{\mathsf{K}}(2)\zeta_{\mathsf{K}}(3)/\zeta_{\mathsf{K}}(6)$.
Since $f_{\mathsf{K}}(s)$ is holomorphic on $\sigma>0$ and
$\zeta_{\mathsf{K}}(s)$ has a unique (simple) pole at $s=1$, we have that
$T_{\mathsf{K}}(s)$ is meromorphic on $\sigma\geq 1$ and has a unique (simple)
pole at $s=1$ with residue

$\displaystyle\mathrm{Res}_{s=1}\Big{[}\,T_{\mathsf{K}}(s)\,\Big{]}=\lim_{s\to
1}(s-1)T_{\mathsf{K}}(s)=\lim_{s\to
1}(s-1)\zeta_{\mathsf{K}}(s)f_{\mathsf{K}}(s)=\rho_{\mathsf{K}}f_{\mathsf{K}}(1)\,.$

As we remarked in Observation 1, the hypothesis of Weiner-Ikehara theorem are
satisfied in this case, so we conclude that

$\displaystyle\lim_{N\to\infty}\dfrac{1}{N}\,\sum_{m=1}^{N}t_{\mathsf{K}}(m)\,=\dfrac{\rho_{\mathsf{K}}\,\zeta_{\mathsf{K}}(2)\zeta_{\mathsf{K}}(3)}{\zeta_{\mathsf{K}}(6)}\,.$

∎

## References

  * [AK13] Allen Altman and Steven Kleiman, _A term of commutative algebra_ , Worldwide Center of Mathematics, 2013.
  * [Bat72] Paul T. Bateman, _The Distribution of Values of the Euler Function_ , Acta Arithmetica 21 (1972), no. 1, 329–345.
  * [Dre70] Robert Dressler, _A Density Which Counts Multiplicity_ , Pacific Journal of Mathematics 34 (1970), no. 2, 371–378.
  * [Erd45] Paul Erdős, _Some remarks on Euler’s $\phi$ function and some related problems_, Bulletin of the American Mathematical Society 51 (1945), no. 8, 540–544.
  * [Ike31] Shikao Ikehara, _An extension of Landau’s theorem in the analytical theory of numbers_ , Journal of Mathematics and Physics 10 (1931), no. 1-4, 1–12.
  * [MV07] Hugh L. Montgomery and Robert C. Vaughan, _Multiplicative number theory i: Classical theory_ , vol. 97, Cambridge University Press, 2007.
  * [Wie32] Norbert Wiener, _Tauberian Theorems_ , Annals of Mathematics (1932), 44–45.


††thanks: These authors contributed equally to this work.††thanks: These
authors contributed equally to this work.

# Probing Multiple Electric Dipole Forbidden Optical Transitions in Highly
Charged Nickel Ions

Shi-Yong Liang  State Key Laboratory of Magnetic Resonance and Atomic and
Molecular Physics, Innovation Academy for Precision Measurement Science and
Technology, Chinese Academy of Sciences, Wuhan 430071, China  Key Laboratory
of Atomic Frequency Standards, Innovation Academy for Precision Measurement
Science and Technology, Chinese Academy of Sciences, Wuhan 430071, China
University of Chinese Academy of Sciences, Beijing 100049, China  Ting-Xian
Zhang  State Key Laboratory of Magnetic Resonance and Atomic and Molecular
Physics, Innovation Academy for Precision Measurement Science and Technology,
Chinese Academy of Sciences, Wuhan 430071, China  University of Chinese
Academy of Sciences, Beijing 100049, China  Hua Guan<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>State Key Laboratory
of Magnetic Resonance and Atomic and Molecular Physics, Innovation Academy for
Precision Measurement Science and Technology, Chinese Academy of Sciences,
Wuhan 430071, China  Key Laboratory of Atomic Frequency Standards, Innovation
Academy for Precision Measurement Science and Technology, Chinese Academy of
Sciences, Wuhan 430071, China  Qi-Feng Lu  Shanghai EBIT Laboratory, Key
Laboratory of Nuclear Physics and Ion-Beam Application (MOE), Institute of
Modern Physics, Fudan University, Shanghai 200433, China  Jun Xiao
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>Shanghai EBIT Laboratory, Key Laboratory of Nuclear Physics and Ion-Beam
Application (MOE), Institute of Modern Physics, Fudan University, Shanghai
200433, China  Shao-Long Chen  State Key Laboratory of Magnetic Resonance and
Atomic and Molecular Physics, Innovation Academy for Precision Measurement
Science and Technology, Chinese Academy of Sciences, Wuhan 430071, China  Key
Laboratory of Atomic Frequency Standards, Innovation Academy for Precision
Measurement Science and Technology, Chinese Academy of Sciences, Wuhan 430071,
China  Max-Planck-Institut für Kernphysik, Heidelberg 69117, Germany  Yao
Huang  State Key Laboratory of Magnetic Resonance and Atomic and Molecular
Physics, Innovation Academy for Precision Measurement Science and Technology,
Chinese Academy of Sciences, Wuhan 430071, China  Key Laboratory of Atomic
Frequency Standards, Innovation Academy for Precision Measurement Science and
Technology, Chinese Academy of Sciences, Wuhan 430071, China  Yong-Hui Zhang
State Key Laboratory of Magnetic Resonance and Atomic and Molecular Physics,
Innovation Academy for Precision Measurement Science and Technology, Chinese
Academy of Sciences, Wuhan 430071, China  Cheng-Bin Li<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>State Key Laboratory
of Magnetic Resonance and Atomic and Molecular Physics, Innovation Academy for
Precision Measurement Science and Technology, Chinese Academy of Sciences,
Wuhan 430071, China  Ya-Ming Zou  Shanghai EBIT Laboratory, Key Laboratory of
Nuclear Physics and Ion-Beam Application (MOE), Institute of Modern Physics,
Fudan University, Shanghai 200433, China  Ji-Guang Li  Institute of Applied
Physics and Computational Mathematics, Beijing 100088, China  Zong-Chao Yan
Department of Physics, University of New Brunswick, Fredericton, New
Brunswick, Canada E3B 5A3  State Key Laboratory of Magnetic Resonance and
Atomic and Molecular Physics, Innovation Academy for Precision Measurement
Science and Technology, Chinese Academy of Sciences, Wuhan 430071, China
Andrei Derevianko  Department of Physics, University of Nevada, Reno, Nevada
89557, USA  Ming-Sheng Zhan  State Key Laboratory of Magnetic Resonance and
Atomic and Molecular Physics, Innovation Academy for Precision Measurement
Science and Technology, Chinese Academy of Sciences, Wuhan 430071, China
Ting-Yun Shi  State Key Laboratory of Magnetic Resonance and Atomic and
Molecular Physics, Innovation Academy for Precision Measurement Science and
Technology, Chinese Academy of Sciences, Wuhan 430071, China  Ke-Lin Gao
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>State Key Laboratory of Magnetic Resonance and Atomic and Molecular Physics,
Innovation Academy for Precision Measurement Science and Technology, Chinese
Academy of Sciences, Wuhan 430071, China  Key Laboratory of Atomic Frequency
Standards, Innovation Academy for Precision Measurement Science and
Technology, Chinese Academy of Sciences, Wuhan 430071, China

###### Abstract

Highly charged ions (HCIs) are promising candidates for the next generation of
atomic clocks, owing to their tightly bound electron cloud, which
significantly suppresses the common environmental disturbances to the quantum
oscillator. Here we propose and pursue an experimental strategy that, while
focusing on various HCIs of a single atomic element, keeps the number of
candidate clock transitions as large as possible. Following this strategy, we
identify four adjacent charge states of nickel HCIs that offer as many as six
optical transitions. Experimentally, we demonstrated the essential capability
of producing these ions in the low-energy compact Shanghai-Wuhan Electron Beam
Ion Trap. We measured the wavelengths of four magnetic-dipole ($M$1) and one
electric-quadrupole ($E$2) clock transitions with an accuracy of several ppm
with a novel calibration method; two of these lines were observed and
characterized for the first time in controlled laboratory settings. Compared
to the earlier determinations, our measurements improved wavelength accuracy
by an order of magnitude. Such measurements are crucial for constraining the
range of laser wavelengths for finding the “needle in a haystack” narrow
lines. In addition, we calculated frequencies and quality factors, evaluated
sensitivity of these six transitions to the hypothetical variation of the
electromagnetic fine structure constant $\alpha$ needed for fundamental
physics applications. We argue that all the six transitions in nickel HCIs
offer intrinsic immunity to all common perturbations of quantum oscillators,
and one of them has the projected fractional frequency uncertainty down to the
remarkable level of 10-19.

##  I INTRODUCTION

Quantum metrology of atomic time-keeping has seen dramatic improvements over
the past decade with novel applications spanning from chronometric geodesy [1,
2] to fundamental physics, such as dark matter searches [3, 4] and multi-
messenger astronomy [5]. Currently, optical atomic clocks using neutral atoms
or singly charged ions have demonstrated fractional frequency uncertainties at
the level of $10^{-18}$ or even $10^{-19}$ [6, 7, 8, 9]. These uncertainties
refer to the ability to protect the quantum oscillator from environmental
perturbations, such as stray magnetic and electric fields. As these existing
technologies mature, they are reaching the stage where one needs to understand
numerous sources of environmental perturbations in greater detail. In some
cases, the perturbations cannot be fully eliminated and one needs to devote
significant efforts to measuring and characterizing the perturbations; these
lead to non-universal systematic corrections to the clock frequency that are
specific to experimental realization of the clock.

Novel classes of atomic clocks must start with quantum oscillators that offer
a much more improved inherent immunity to environmental perturbations than the
more mature technologies. One of such systems is the nuclear clock based on
the unique property of the 229Th nucleus – the existence of a nuclear
transition in a laser-accessible range [10, 11]; unfortunately, despite
substantial world-wide efforts [12, 13], this transition is yet to be observed
directly. The suppression of environmental perturbations for the nuclear
oscillator comes from the nuclear size being $\sim 10^{4}$ times smaller than
the size of a neutral atom. Alternative novel systems are highly charged ions
(HCIs) [14, 15]. Similar to the nuclear clock, here the oscillator size is
also substantially reduced, owing to the electronic cloud size shrinking as
$1/Z$ with the increasing ion charge $Z$. HCIs were proposed as promising
candidates for the next generation of atomic clocks [14]. In addition, beyond
the improved time-keeping, HCIs open intriguing opportunities for probing new
physics beyond the standard model of particle physics [16, 17].

Compared to the single, yet to be spectroscopically found nuclear transition,
there is a plethora of suitable HCIs (see a review [18] for a sample of
proposals). A detailed analysis [19] indicates that with certain HCIs, atomic
clocks “can have projected fractional accuracies beyond the
$10^{-20}-10^{-21}$ level for all common systematic effects, such as blackbody
radiation, Zeeman, ac-Stark, and quadrupolar shifts”. Moreover, compared to
the nuclear clock, where the direct observation of the clock transition
remains elusive, the clock transitions in HCIs can be found with conventional
spectroscopy or from atomic-structure computations. Indeed, here we report
spectrographic measurements of wavelengths for five clock transitions with an
accuracy of several ppm (see Table 1), setting up the stage for the more
accurate laser spectroscopy.

Table 1:  Observed and calculated wavelengths of magnetic-dipole ($M$1) and electric-quadrupole ($E$2) transitions, where the lines $a$ through $f$ are candidate clock transitions, in nm.  Line | Ion | Transition | Type | NIST | This work
---|---|---|---|---|---
Vacuum | Air (observed) | Vacuum | Theory
$a$ | Ni11+ | $3s^{2}3p^{5}\;\;{}^{2}\\!P_{1/2}-{{}^{2}\\!P}_{3/2}$ |  $M$1 | 423.2 | 423.104(2) | 423.223(2) | 423.0(6)
$b$ | Ni12+ | $3s^{2}3p^{4}\;\;{}^{3}\\!P_{1}-{{}^{3}\\!P}_{2}$ |  $M$1 | 511.724 | 511.570(2) | 511.713(2) | 511.8(6)
$c$ | Ni14+ | $3s^{2}3p^{2}\;\;{}^{3}\\!P_{1}-{{}^{3}\\!P}_{0}$ |  $M$1 | 670.36 | 670.167(2) | 670.352(2) | 671.1(14)
$d$ | Ni15+ | $3s^{2}3p\;\;^{2}\\!P_{3/2}-{{}^{2}\\!P}_{1/2}$ |  $M$1 | 360.22 | 360.105(2) | 360.207(2) | 359.9(9)
$e$ | Ni12+ | $3s^{2}3p^{4}\;\;{}^{3}\\!P_{0}-{{}^{3}\\!P}_{2}$ |  $E$2 | 498.50(249) | – | – | 496.9(24)
$f$ | Ni14+ | $3s^{2}3p^{2}\;\;{}^{3}\\!P_{2}-{{}^{3}\\!P}_{0}$ |  $E$2 | 365.277 | – | 365.278(1) | 365.0(3)
$g$ | Ni14+ | $3s^{2}3p^{2}\;\;{}^{3}\\!P_{2}-{{}^{3}\\!P}_{1}$ |  $M$1 | 802.63 | 802.419(2) | 802.639(2) | 800.3(25)

Despite the lack of suitable electric-dipole ($E$1) transitions for direct
laser cooling, recent successes in sympathetic cooling and quantum logic
spectroscopy of HCIs have paved way for precision spectroscopic measurements
with HCIs [20, 21]. It is worth emphasizing that these newly demonstrated
technologies can be applied universally to a wide range of HCIs. The multitude
of suitable clock HCI candidates is a “blessing in disguise”, as one needs to
commit to building the infrastructure for a specific ion. As with any new
endeavor, one would like to mitigate potential problems with picking a “wrong”
ion. Here we propose and pursue a straddling strategy that would allow one to
explore several clock transitions using not only the same HCI production
system but also ions of the same atomic element.

A suitable HCI has to possess a number of properties enabling precision
spectroscopy and compatibility with operating an atomic clock. Generally, one
may distinguish between three classes of visible or near visible optical
forbidden transitions in HCIs that can be used for developing optical clocks:

  1. 1.

Magnetic-dipole ($M$1) transitions between two hyperfine-structure levels of
the same electronic state [22, 19].

  2. 2.

Forbidden transitions between level crossing electronic states, which tend to
be sensitive to variation of the fine structure constant [16, 23, 24, 25].

  3. 3.

Forbidden transitions between the ground-state fine structure levels [14, 19,
26, 27].

Type 1 transitions occur in few-electron heavy HCIs [19] that are challenging
to produce and trap. Type 2 transitions involve a complex energy structure
that can impede initialization and read-out of the clock states. Here we focus
on type 3 transitions that offer simplicity in both producing the ions and
clock operation. More specifically we choose HCIs of nickel of various charge
states [19, 26, 27]: Ni11+, Ni12+, Ni14+, and Ni15+. The clock transitions are
shown in Fig. 1. All the traditional clock perturbations are strongly
suppressed for these ions due to the charge scaling arguments [28, 14, 19]. As
pointed out in Ref. [14], the major issue with HCI clocks is the so-called
quadrupolar shift of the clock transition, when the quadrupole ($Q$) moment of
the clock state couples to the always existing $E$-field gradients in ion
traps. While the $Q$-moment of an electronic cloud does scale as $1/Z^{2}$,
this reduction is not sufficient to suppress the quadrupolar shift below the
desired level of accuracy. Thus, it is beneficial to select clock states with
either vanishing or strongly suppressed $Q$-moments.

There are four $M$1 and two $E$2 optical transitions in Ni11+, Ni12+, Ni14+,
and Ni15+ that offer the desired flexibility. These ions have varying number
of electrons in the $3p$ shell, see Fig. 1. The clock transitions are between
the fine structure components of the ground electronic state. There are four
stable isotopes 58Ni, 60Ni, 62Ni, and 64Ni without nuclear spin; these can be
used to search for new physics with isotope shift measurements [29, 30, 31]
and for initial spectroscopic measurements. These spin-0 isotopes, however,
will be susceptible to the quadrupolar shifts for clock transitions. However,
these shifts can be suppressed by using the 61Ni isotope (nuclear spin
$I=3/2$), which has a natural abundance of 1.14%. Then the quadrupolar shifts
can be either strongly suppressed or completely removed by employing the
following clock transitions between hyperfine states (see Fig. 1):

  * •

${{}^{2}\\!P}_{3/2}\,F=0$ and ${}^{2}\\!P_{1/2}\,F=1$ or $F=2$ for Ni11+ and
Ni15+,

  * •

${}^{3}\\!P_{1}\,F=1/2$ and ${}^{3}\\!P_{2}\,F=1/2$ for Ni12+,

  * •

${}^{3}\\!P_{1}\,F=1/2$ and ${}^{3}\\!P_{0}\,F=3/2$ for Ni14+,

  * •

${}^{3}\\!P_{0}\,F=3/2$ and ${}^{3}\\!P_{2}\,F=1/2$ for Ni12+ and Ni14+.

This selection is based on the following reasoning [19]: $Q$-moments (rank 2
tensors) of the $F=0$ and $F=1/2$ states are zero due to selection rules. For
the ${}^{2}\\!P_{1/2}\,F=1,2$ and ${}^{3}\\!P_{0}\,F=3/2$ states, the
electronic $Q$-moments vanish due to selection rules for the electronic
angular momentum $J$. Thereby, the $Q$-moments are determined by the nuclear
$Q$-moments or hyperfine mixing [11] and, as such, are strongly suppressed. As
an indication of attainable accuracy, Refs. [26, 27] evaluated relevant
properties of the clock transitions in 61Ni15+ and 58Ni12+ and estimated
common systematic uncertainties to be below $10^{-19}$, in line with the more
general estimates of Ref. [19]. The second-order Doppler shift induced by the
excess micromotion of the trapped ion is expected to be suppressed to below
$10^{-19}$ by compensating the stray electric field to a level below 0.1 V/m
[32, 33]. In a cryogenic trap, the heating rate of the trapped ions caused by
the collisions with the background gas and the anomalous motional heating is
reduced, and hence the second-order Doppler shift induced by the secular
motion is also expected to be sufficiently small [18]. Based on these
arguments, we expect the attainable fractional systematic uncertainty of all
the six clock transitions in Ni HCIs to be $10^{-19}$.

Figure 1:  Partial energy-level diagrams for highly charged nickel ions. Clock
transitions are explicitly drawn. Magnetic-dipole ($M$1) transitions are shown
in magenta and electric-quadrupole ($E$2) transitions in green. The labeling
of transitions is the same as in Table 1.

As the first essential step towards realizing the Ni HCI clocks, we produced
the target ions at our newly built low-energy compact Shanghai-Wuhan Electron
Beam Ion Trap (SW-EBIT) [34]. The wavelengths of four $M$1 and one $E$2 clock
transitions between the ground-state fine structure levels in these ions are
measured to an accuracy of several ppm using a spectrograph. In particular,
the three $M$1 lines $b$, $c$, and $g$ (listed in Table 1) in Ni12+ and Ni14+
are observed and characterized for the first time in the laboratory. We also
carried out calculations for these ions using an ab initio relativistic method
of atomic structure, the multi-configuration Dirac-Hartree-Fock (MCDHF) method
[35, 36]. We evaluated relevant spectroscopic properties, such as transition
wavelengths and natural linewidths. We also estimated the sensitivity to the
hypothetical variation of the fine structure constant $\alpha$ and found that
all considered clock transitions in Ni HCIs are more susceptible to the
variation than most of the commonly employed singly charge ions or neutral
atoms. Thus, Ni HCIs can be used for placing stringent constraints on the
spatial or temporal variation of $\alpha$.

##  II EXPERIMENTAL METHOD AND RESULTS

###  II.1 Production of Ni HCIs

To produce Ni HCIs, we injected the Ni(C5H5)2 (nickelocene) molecular beam
into the trap center. Then the charge-state distribution of Ni HCIs was
measured using the electron-beam current of 6 mA and the electron-beam energy
of 500 eV, which is higher than the ionization energies 319.5 eV, 351.6 eV,
429.3 eV, and 462.8 eV needed for Ni11+, Ni12+, Ni14+, and Ni15+,
respectively. The extraction period was 0.3 s and the magnetic flux density
was 0.16 T. As shown in Fig. 2, the target ions Ni11+, Ni12+, Ni14+, and Ni15+
were produced, and the ions of two distinct isotopes, 60Ni and 58Ni, were
resolved. The techniques for measuring charge-state distribution are described
in Ref. [34].

Figure 2:  Charge-state distribution of the Ni HCIs, obtained by averaging 3
measurements.

###  II.2  Spectral measurements

The spectra of the trapped Ni HCIs were observed by a Czerny-Turner
spectrograph (Andor Kymera 328i) equipped with an Electron Multiplying Charge-
Coupled Device (EMCCD, Andor Newton 970, pixel: $1600\times 200$, pixel size:
16 $\mu$m) and a 1200 l/mm grating blazed at 500 nm. To maximize the number of
the Ni HCIs of a specific charge state, different electron-beam energies were
used, i.e. 370 eV, 400 eV, 500 eV, and 540 eV for Ni11+, Ni12+, Ni14+, and
Ni15+, respectively. As illustrated in Fig. 3, the fluorescence emitted from
the Ni HCIs was focused by a single N-BK7 Bi-Convex lens (focal length $f=10$
cm at 633 nm) on the spectrograph entrance slit. The distance between the trap
(DT2, drift tube 2 in SW-EBIT [34]) center and the front principal plane of
the lens remained fixed at 197 mm, which was about twice the focal length.
Before setting up the spectrograph, a Charge-Coupled Device (CCD) was placed
on the image plane to image the two inner edges of DT2 (1 mm slit width) that
were illuminated by the hot cathode. In order to ensure that the lens was
aligned with the optical axis, we adjusted the angle and position of the lens
until the edge image became mirror-symmetric. Because of the dispersion of the
lens, to ensure that the spectrograph slit was always precisely located on the
image plane, the distance $L$ between the slit of the spectrograph and the
back principal plane of the lens was calculated and adjusted for every central
wavelength of the measured spectra. The grating was set to zero-order to image
the inner edges of DT2 through spectrograph with its maximum slit width and
minimum iris aperture behind the slit. Similarly, the angle and position of
the spectrograph were adjusted until the image of the edges became mirror-
symmetric to ensure the spectrograph alignment with the optical axis. A one
inch aperture was placed before the lens to block the stray light. For
calibration, a conjugated optical system was installed on the opposite side of
the spectrograph. A diffuser attached by a 0.5 mm slit was placed on the
object plane. A low-pressure gas-discharge lamp filled with Kr illuminated the
slit, and the slit was imaged to the trap center to overlap with the trapped
ion cloud. During the spectral exposure time of 10 to 60 minutes, the Kr lamp
as the calibration light source flashed at a period of 1 to 3 minutes. The
slit of the spectrograph was set to 30 $\mu$m, and the iris aperture in the
spectrograph was set to 40 steps to obtain the F/7.6 aperture. The focal
length of the spectrograph was tuned to minimize the linewidth in each
spectral range.

All the spectra were binned to a non-dispersive direction after removing the
cosmic ray noise, as shown in Fig. 4 (a). The dispersion function was obtained
by fitting the NIST-tabulated Ritz in-the-air wavelengths of the calibration
lines to a cubic polynomial, against their column numbers of the line
centroids. The residuals of the calibration lines and the 1-$\sigma$ fitting
confidence band are shown in Fig. 4 (b). To determine the line centroids, the
measured lines and calibration lines were fitted to a Gaussian or a multi-
Gaussian profile, as shown in Fig. 4 (c).

Figure 3:  Scheme of observation and calibration of the measured lines. The Ni
HCIs are trapped at the center of DT2 in SW-EBIT.  Figure 4:  (a) A spectrum
of line $a$ from Ni11+ and its calibration lines from Kr atom whose
approximate wavelengths are labeled in the figure in nm. (b) Residuals of
cubic polynomial fits of the calibration lines. The gray band is a 1-$\sigma$
confidence band. The uncertainties in the calibration lines are dominated by
the line centroid uncertainties of the Gaussian fits. (c) Spectrum of line $a$
and its Gaussian fit.

###  II.3 Observed wavelengths

Previously, these five $M$1 lines in Table 1 were observed in the solar corona
emission [37] with a wavelength uncertainty of tens of picometer. The lines
$a$ and $d$ have been also measured in Tokamak [38, 39], but experimental
observation of the other three lines $b$, $c$, and $g$ has not been reported
in the literature. In this work, we observed and identified all five $M$1
lines emitted from the nickel plasma in the SW-EBIT in a controlled laboratory
setting. The measured wavelengths agree with the Ritz wavelengths in NIST
database [40], as shown in Table 1, where the wavelengths between air and
vacuum were converted by an empirical equation [41]. However, for line $d$,
our result of 360.105(2) nm substantially deviates from the value of
360.123(2) nm observed from Tokamak plasma by Hinnov  _et al._ [39]. To test
our result, two lines from Ar+ were measured without any change of the optical
system comparing to the measurement of line $d$, and the measured wavelengths
in air were 357.660(2) nm and 358.843(2) nm, which were in good agreement with
the Ritz wavelengths in NIST database, i.e., 357.661538 nm and 358.844021 nm.
For the $E$2 lines $e$ and $f$, the transition rates are too small to be
observable by our technique. However, we deduced the wavelength of line $f$ in
Table 1 from those of lines $c$ and $g$ via the Rydberg-Ritz combination
principle.

###  II.4 Measurement uncertainties

Line centroid uncertainty. The line centroids of the measured lines and their
calibration lines were determined by the centers of the fitted Gaussian
profiles. Since the statistical uncertainty of the line centroid was mainly
caused by the low signal-to-noise ratio, we evaluated the statistical
uncertainty by performing at least 20 measurements on the line of interest, as
shown in Fig 5. For all five measured lines, this uncertainty was smaller than
0.4 pm. The systematic uncertainty of the line centroid is mainly caused by
the non-ideal Gaussianity of the line because of the optical aberration and
the Zeeman components. In this work, since the measured lines and their
calibration lines shared a similar profile, the optical aberration effect was
largely offset. In the trap center, the magnetic flux density was $\sim$0.16
T, resulting in a $\sim$2 pm splitting between the Zeeman components of the
clock transitions, which was relatively small (unresolved) compared to the
$\sim$90 pm linewidth. Furthermore, the Zeeman effect would not alter the line
centroid because the Zeeman components were symmetrically distributed; in
addition, the Zeeman effect was negligible for the Kr lamp due to the low
magnetic field of 0.4 mT.

Figure 5:  The calibrated wavelengths of line $a$ in air derived from a series
of 26 measurements. The wavelength uncertainty of each single spectrum was
calculated as the quadrature of the line centroid uncertainty and the
1-$\sigma$ confidence interval of the fitted dispersion function. The weighted
average wavelength is represented by the solid purple line and its uncertainty
is represented by lilac band.

Dispersion function uncertainty. The statistical uncertainty for the
dispersion function was caused by the centroid statistical uncertainties of
calibration lines, which were reduced by the statistics of the line centroids.
The systematic dispersion function uncertainty of a measured line was
estimated by averaging the absolute values of the fitted residuals of its
calibration lines of all the measured spectra.

Calibration systematic uncertainty. Since the image of the calibration light
source might not be overlapped exactly with the trapped ion cloud, the spatial
deviation and misalignment could cause wavelength offset between the measured
lines and their calibration lines. In this work, a spatial deviation of less
than 2 mm would result in a wavelength uncertainty of less than 1 pm. The
misalignment could cause a wavelength uncertainty of less than 1 pm, which was
estimated from five measurements of the Ar9+ 553 nm line by resetting the
optical system every time. Thereby, the overall systematic uncertainty caused
by our calibration scheme was expected to be less than 2 pm.

Other uncertainties. In this work, the calibration light source and the
fluorescence of the trapped ions were exposed to the spectrograph almost
simultaneously, indicating that the temperature drift and the mechanical drift
were canceled out. The shifts due to the Stark effect and collisions can also
be neglected at this level of accuracy. The wavelengths of the selected
calibration lines are reliable because their uncertainties in the NIST
database are all less than 0.3 pm.

Table 2 is the uncertainty budget for the lines $a$-$d$ and $g$ in air. The
total uncertainty was calculated as the quadrature of all the uncertainties,
which was dominated by the calibration systematic uncertainty. In order to
test the reliability of the uncertainty estimation, the wavelengths in-the-air
of the three lines from Ar HCI were measured, i.e., Ar9+ 553.327(2) nm, Ar10+
691.689(2) nm, and Ar13+ 441.255(2) nm, which were consistent with the
previous measured values of 553.3265(2) nm, 691.6878(12) nm, and 441.255919(6)
nm, respectively [42, 43].

The total wavelength uncertainty of this observation and calibration scheme
was approximately 2 pm, which was comparable to the uncertainty of the scheme
that the measured lines were calibrated by the lines from the buffer gas
observed by a similar resolution spectrograph [44, 45], but larger than the
uncertainty of the scheme that the calibration source was overlapped with the
real image of the ion cloud observed by a higher resolution spectrograph [42].
Compared to these two schemes, our scheme is more convenient and flexible. The
uncertainty may be reduced by using a higher resolution spectrograph that is
less sensitive to the calibration optical system.

Table 2:  Uncertainty budget of the measured lines.  Source | Uncertainty in wavelength (pm)
---|---
Line |  $a$ |  $b$ |  $c$ |  $d$ |  $g$
Line centroid |  0.2 |  0.2 |  0.1 |  0.2 |  0.3
Dispersion function |  0.1 |  0.3 |  0.3 |  0.2 |  0.4
Calibration systematic |  2 |  2 |  2 |  2 |  2
Total |  2 |  2 |  2 |  2 |  2

##  III THEORETICAL METHOD AND RESULTS

###  III.1  MCDHF calculations

In the MCDHF method, an atomic wave function $\Psi$ is constructed as a linear
combination of configuration state functions (CSFs) $\Phi$ of the same parity
$P$, the total angular momentum $J$, and its projection $M_{J}$, i.e.,

$\Psi(\Gamma
PJM_{J})=\sum_{i=1}^{N_{\mathrm{CSF}}}{c_{i}\Phi(\gamma_{i}PJM_{J})}.$ (1)

Here $c_{i}$ is the mixing coefficient and $\gamma_{i}$ stands for the
remaining quantum numbers of the CSFs. Each CSF itself is a linear combination
of products of one-electron Dirac orbitals. Both mixing coefficients and
orbitals are optimized in the self-consistent field calculation. After a set
of orbitals is obtained, the relativistic configuration interaction (RCI)
calculations are used to capture more electron correlations. In addition to
the Coulomb interactions, our RCI calculations also include the Breit
interaction in the low-frequency approximation and the quantum electrodynamic
(QED) corrections.

In order to obtain high-quality atomic wave functions, we designed an
elaborate computational model as follows. In the first step, the self-
consistent field (SCF) calculations were performed successively to generate
virtual orbitals. The virtual orbitals were employed to form CSFs which
account for certain electron correlations. More specifically, CSFs were
produced through single (S)- and double (D)-electron excitations from the
occupied Dirac-Hartree-Fock orbitals to virtual orbitals, but the double
excitation from the atomic core $1s^{2}2s^{2}2p^{6}$ orbitals were excluded at
this stage. The virtual orbitals were augmented layer by layer up to
$n_{\rm{max}}=12$ and $l_{\rm{max}}=6$, where $n_{\rm{max}}$ and
$l_{\rm{max}}$ denote, respectively, the maximum principal quantum number and
the maximum angular quantum number of the virtual orbitals. In the second
step, the single-reference configuration RCI calculations were performed with
the configuration space constructed from SD excitation from all occupied
orbitals to the set of virtual orbitals. In other words, the correlation
between electrons in the atomic core, which were neglected in the first step,
were captured. In the last step, we considered part of contributions from the
triple- and quadruple-excitation CSFs. In order to limit the number of CSFs,
the MR-SD approach was adopted to produce corresponding CSFs [46, 47]. The
multi-reference (MR) configuration sets were created as $\\{3s3p^{5}3d$,
$3s^{2}3p^{3}3d^{2}\\}$ for Ni11+, $\\{3s3p^{4}3d$, $3s^{2}3p^{2}3d^{2}$,
$3p^{6}\\}$ for Ni12+, $\\{3s3p^{2}3d$, $3s^{2}3d^{2}$, $3p^{4}\\}$ for Ni14+,
and $\\{3p^{3}$, $3s3p3d$, $3p3d^{2}\\}$ for Ni15+. Additionally, the Breit
interaction and the QED corrections were included in the RCI computation.

Once the atomic wave functions are obtained, we are in a position to calculate
the physical quantities under investigation, that is, the reduced matrix
elements for corresponding rank $k$ irreducible tensor operators between two
atomic states, i.e., $\langle\Psi(\Gamma
PJ)\|O^{(k)}\|\Psi(\Gamma^{{}^{\prime}}P^{{}^{\prime}}J^{{}^{\prime}})\rangle\,.$
The magnetic dipole and electric quadrupole transition operators are rank 1
and rank 2 operators, respectively. In practice, we performed the calculations
using the GRASP2018 package [48].

###  III.2 Calculated wavelengths

As shown in Table 1, the calculated wavelengths of the $M$1 transitions of
line $a$ through line $d$ and line $g$ agree with our measured values. The
wavelengths of the two $E$2 transitions of line $e$ and line $f$ in Ni12+ and
Ni14+ were also calculated. These two lines have not been observed yet before.
Our calculated wavelengths for these two transitions are in agreement with the
NIST recommended values. Meanwhile, the calculated wavelength of line $f$ also
agrees with our indirect measurement, see Table 1.

###  III.3 Properties of the clock transitions

The design of an atomic clock relies on the knowledge of atomic parameters of
the quantum oscillator. Thus, we have computed wavelengths, spontaneous
emission rates $A$, lifetimes $\tau$, linewidths $\Gamma$
($2\pi\Gamma=1/\tau$), and other parameters for all six candidate clock
transitions, and the results are listed in Table 3. As one of the key
parameters of clock stability, the quality factor ($Q$-factor) is also given
in this table. The $Q$-factor is defined as the ratio of the clock frequency
$\nu_{\mathrm{clk}}$ to the linewidth $\Gamma$ of the clock transition, i.e.,
$Q=\nu_{\mathrm{clk}}/\Gamma$. Among the four $M$1 clock transitions, the
${}^{3}\\!P_{1}-{{}^{3}\\!P}_{0}$ transition in Ni14+ is the narrowest with
its linewidth less than $10\,\mathrm{Hz}$, while the linewidths of the other
three $M$1 transitions are about $30\,\mathrm{Hz}$. The corresponding
$Q$-factors of these four $M$1 transitions are $\sim 10^{13}$. There are two
decay channels from ${}^{3}\\!P_{0}$ in Ni12+ and ${}^{3}\\!P_{2}$ in Ni14+ to
the lower states. In order to determine the linewidth of these $E$2-clock
transitions , both decay channels should be taken into account. For
${}^{3}\\!P_{0}$ in Ni12+, the decay rate is 0.037 s-1 for the $E$2
(${}^{3}\\!P_{0}-{{}^{3}\\!P}_{2}$) channel and 0.011 s-1 for the $M$1
(${}^{3}\\!P_{0}-{{}^{3}\\!P}_{1}$) channel. For ${}^{3}\\!P_{2}$ of Ni14+,
the $E$2 (${}^{3}\\!P_{2}-{{}^{3}\\!P}_{0}$) and $M$1
(${}^{3}\\!P_{2}-{{}^{3}\\!P}_{1}$) transition rates are 0.03 s-1 and 22.5
s-1, respectively. Therefore, the linewidths for the $E$2-clock transitions
are $3.6~{}\mathrm{Hz}$ for ${}^{3}\\!P_{2}-{{}^{3}\\!P}_{0}$ in Ni14+ and
$8~{}\mathrm{mHz}$ for ${}^{3}\\!P_{0}-{{}^{3}\\!P}_{2}$ in Ni12+, which are
respectively smaller than the $M$1 transition lines c and b, as marked in Fig.
1. This $E$2 transition in Ni12+ is particularly attractive for stable
clockwork [27], because of its relatively high $Q$-factor of $7.5\times
10^{16}$, meaning that the statistical uncertainty limited by the quantum
projection noise [49, 50, 18] of this transition can reach the level of
$10^{-19}$ by averaging over a few days.

Table 3:  Theoretical spectral properties of clock transitions. Here $A$ is the Einstein coefficient for spontaneous decay, $\tau$ is the lifetime of the upper clock state, $\Gamma$ is the natural linewidth, and $Q$ is the transition quality factor. Also, $q$ and $K$ are, respectively, the sensitivity coefficient and enhancement factor for the variation of the fine structure constant. Numbers in square brackets stand for the powers of 10, i.e., $x[y]\equiv x\times 10^{y}$.  Transition | Type |  $A$ (s-1) |  $\tau$ (ms) |  $\Gamma$ (Hz) | $Q$ |  $q$ (cm-1) | $K$ |
---|---|---|---|---|---|---|---|---
Ni11+ $3s^{2}3p^{5}$
${}^{2}\\!P_{1/2}-{{}^{2}\\!P}_{3/2}$ |  $M$1 | 238(2) | 4.2(1) | 38 | 1.9[13] | 24820 | 2.1 |
|  | 236.31(3) | 4.23(2) |  |  | 24464 |  | Ref. [51]
Ni12+ $3s^{2}3p^{4}$
${}^{3}\\!P_{1}-{{}^{3}\\!P}_{2}$ |  $M$1 | 157(1) | 6.3(1) | 25 | 2.3[13] | 22473 | 2.3 |
|  | 154 | 6.5 |  |  |  |  | Ref. [27]
${}^{3}\\!P_{1}-{{}^{3}\\!P}_{2}$ |  $E$2 | 0.02 |  |  |  |  |  |
${}^{3}\\!P_{0}-{{}^{3}\\!P}_{2}$ |  $E$2 | 0.037(4) | 21(3)[3] | 0.008 | 7.5[16] | 14982 | 1.5 |
|  | 0.03 | 19[3] |  | 1.1[16] |  |  | Ref. [27]
${}^{3}\\!P_{0}-{{}^{3}\\!P}_{1}$ |  $M$1 | 0.011(2) |  |  |  |  |  |
Ni14+ $3s^{2}3p^{2}$
${}^{3}\\!P_{1}-{{}^{3}\\!P}_{0}$ |  $M$1 | 56.1(5) | 17.8(2) | 9 | 5.0[13] | 20340 | 2.7 |
${}^{3}\\!P_{2}-{{}^{3}\\!P}_{0}$ |  $E$2 | 0.030(1) | 44(1) | 3.6 | 2.3[14] | 28197 | 2.1 |
${}^{3}\\!P_{2}-{{}^{3}\\!P}_{1}$ |  $M$1 | 22.5(4) |  |  |  |  |  |
${}^{3}\\!P_{2}-{{}^{3}\\!P}_{1}$ |  $E$2 | 0.001 |  |  |  |  |  |
Ni15+ $3s^{2}3p$
${}^{2}\\!P_{3/2}-{{}^{2}\\!P}_{1/2}$ |  $M$1 | 193(2) | 5.2(1) | 31 | 2.7[13] | 29204 | 2.1 |
|  | 190.99 | 5 | 30.38 | 2.73[13] | 89391 |  | Ref. [26]

From the perspective of searching for new physics, we anticipate that by
monitoring the Ni HCI clock transition frequencies, stringent constraints
could be placed on the possible time variation of the fine structure constant
$\alpha$. Following Refs. [52, 17], one can introduce the “sensitivity
coefficient” $q$, defined by $\omega(x)=\omega_{0}+qx$, where
$x\equiv(\alpha/\alpha_{0})^{2}-1$ and $\omega_{0}$ is the clock transition
frequency at the nominal value of the fine structure constant $\alpha_{0}$.
The sensitivity coefficient $q$ characterizes the linear response of the clock
frequency $\omega(x)$ to the variation of $\alpha$, and can be calculated
numerically as $q\approx[\omega(+x)-\omega(-x)]/{(2x)}$. Another commonly used
quantity is the dimensionless enhancement factor [17]
$K=\partial\ln\omega/\partial\ln\alpha\approx 2q/\omega_{0}$. As shown in
Table 3, our computed $K$ values for the relevant transitions in nickel HCIs
are about $2$, which is higher than most of the current optical clocks. For
example [53], Al+ has $K=0.008$. Out of $\sim 10$ species currently used in
the optical clock community, only the heavy Yb+ and Hg+ ions have $|K|>2$
[53]. Therefore, we expect that, even with their initial predicted accuracy of
$10^{-19}$, the quantum clocks based on the relatively light Ni HCIs will have
greater potential for exploring new physics than most of the current atomic
clocks. Recently, an improved constraint of
$\dot{\alpha}/\alpha=1.0(1.1)\times 10^{-18}/$year was reported based on the
comparison of the ${}^{2}S_{1/2}(F=0)-{{}^{2}D}_{3/2}(F=2)$ ($E$2, $K=1.00$)
and the ${}^{2}S_{1/2}(F=0)-{{}^{2}F}_{7/2}(F=3)$ ($E$3, $K=-5.95$) transition
of 171Yb+ clock [54]. The constraint on the temporal variation of $\alpha$ is
expected to be further improved by comparing two clocks based on the $E$2
transition of Ni12+ and the $E$3 transition of Yb+, because of its larger $K$
value and smaller projected systematic and statistical uncertainties of the
$E$2 transition in Ni12+ than those of the $E$2 transition in Yb+.

Nandy and Sahoo [51] determined the sensitivity coefficient to the
$\alpha$-variation for the ${{}^{2}\\!P_{1/2}}-{{}^{2}\\!P_{3/2}}$ transition
in Ni11+ ion. In their work, the transition rate and the lifetime of the
${{}^{2}\\!P_{1/2}}$ state were calculated using the relativistic coupled-
cluster (RCC) method. Yu and Sahoo [26, 27] calculated some atomic parameters
for the ${{}^{2}\\!P_{3/2}}-{{}^{2}\\!P_{1/2}}$ transition in Ni15+ and the
${{}^{3}\\!P_{0}}-{{}^{3}\\!P_{2}}$ transition in Ni12+ with the RCC and MCDHF
methods. Their results are also listed in Table 3 for comparison. For lines
$a$, $d$, and $e$, our calculated values agree well with other theoretical
results [51, 26, 27], except for a factor of 3 difference for the sensitivity
coefficient $q$ of line $d$. There is also a factor of 6 difference in the
value of the $Q$-factor of line $e$, for which we traced back to the trivial
factor of $2\pi$ missing in the linewidth definition in Ref. [27].

Previous theoretical work on nickel HCIs focuses on atomic properties relevant
to the emission from the solar, astrophysical, and laboratory plasmas. In
Tables 4 and 5, we present a comparison with the literature values for the
spontaneous decay rates and lifetimes. Overall, our MCDHF values agree well
with the results from other theoretical methods, such as the RCC method and
the multi-reference Møller-Plesset perturbation theory. Moreover, the
lifetimes of the ${}^{2}\\!P_{1/2}$ state in Ni11+ ion, the ${}^{3}\\!P_{1}$
state in Ni12+ ion, and the ${}^{2}\\!P_{3/2}$ state in Ni15+ ion were
measured at the heavy-ion storage ring [55, 56, 57]. We found a good agreement
between theory and experiment.

Table 4:  Spontaneous emission rates in Ni HCIs, in s-1.  Line | This work | Other theory
---|---|---
$a$ | 238(2) | 235 [58], 260 [59], 236.31(3) [51], 237 [60], 213.1 [61]
$b$ | 157(1) | 154 [27],156.9 [62], 157.4 [63, 64], 157 [60, 65], 156 [66]
$c$ | 56.1(5) | 56.08 [67], 57 [68], 52.7 [69], 56.45 [62], 56.42 [70], 56.5 [60], 54.66 [71], 56 [66]
$d$ | 193(2) | 192.2 [72], 190.99 [26]
$e$ | 0.037(4) | 0.034 [27], 0.03622 [62] 0.037 [64],0.03702 [63], 0.0355 [65], 0.048 [66]
$f$ | 0.030(1) | 0.03 [67],0.029 [69], 0.03044 [62], 0.0157 [70], 0.031 [71] 0.028 [66]
Table 5:  Lifetimes (in ms) of upper clock states in Ni11+, Ni12+, Ni14+, and Ni15+. Numbers in square brackets stand for the powers of 10, i.e., $x[y]\equiv x\times 10^{y}$.  Ion | State | This work | Other theory | Experiment
---|---|---|---|---
Ni11+ | ${{}^{2}\\!P_{1/2}}$ | 4.2(1) | 4.25 [58], 4.23(2) [51] | 4.166(60) [55]
Ni12+ | ${}^{3}\\!P_{1}$ | 6.3(1) | 6.5 [27], 6.55 [73], 6.59 [73] | 7.3(2)∗, 6.50(15)∗∗ [56]
Ni12+ | ${}^{3}\\!P_{0}$ | 21(3)[3] | 22.1[3] [73], 19.5[3] [73], 19[3] [27] |
Ni14+ | ${}^{3}\\!P_{1}$ | 17.8(2) | 17.8 [67], 17.7 [70] |
Ni14+ | ${}^{3}\\!P_{2}$ | 44(1) | 44.6 [67], 45.1 [70] |
Ni15+ | ${{}^{2}\\!P_{3/2}}$ | 5.2(1) | 5.2 [72], 5 [26], 5.184 [74] | 5.90(1)∗, 5.27(7)∗∗ [57]
* Single exponential evaluation
** Multi-exponential evaluation

###  III.4 Computational uncertainties

The computational uncertainties in our work include the neglected correlation
contributions, such as the triple- and quadruple-electron excitations
involving the $1s$ orbital. The upper limit on these effects was estimated
from the double excitations of the core orbitals in the single-reference
configuration RCI calculations. The “truncation” uncertainties due to the
finite number of virtual orbitals were evaluated based on the convergence
trends in the above-mentioned three steps. For the wavelengths, all the
uncertainties were summed together in quadrature. For the $M$1 transition
rates, in addition to these uncertainties, we also included the frequency-
dependent Breit interaction contribution as another source of error. For the
$E$2 transition rates, the difference in results between the Babushkin and
Coulomb gauges [75] were treated as an additional contribution to the combined
theoretical uncertainty.

##  IV CONCLUSIONS

To reiterate, the quantum clockwork we explored here provides an intriguing
possibility for achieving high accuracy on multiple transitions in HCIs of the
same element. Our strategy offers an important flexibility in the pursuit of
multiple candidate clock transitions. Particularly, the $E$2 transition in
61Ni12+ has projected fractional uncertainty $10^{-19}$. We demonstrated the
key experimental capabilities of using our SW-EBIT facility to generate and
extract Ni11+, Ni12+, Ni14+, and Ni15+ ions. We measured the wavelengths of
four $M$1 and one $E$2 clock transitions in these ions with the uncertainties
of about 2 pm. The measured wavelengths establish an important reference for
precision laser spectroscopy in future clock transition measurements. We also
calculated spectroscopic properties of the relevant $M$1 and $E$2 clock
transitions. The calculated wavelengths are consistent with our experimental
results and with previous determinations. The calculated properties indicate
that these ions are suitable for precision quantum metrology and for exploring
new physics beyond the standard model of particle physics.

###### Acknowledgements.

The authors thank Xin Tong, José R. Crespo López-Urrutia, and Yan-Mei Yu for
helps and for fruitful discussions. This work was supported by the Strategic
Priority Research Program of the Chinese Academy of Sciences (Grant No.
XDB21030300), the National Natural Science Foundation of China (Grant Nos.
11934014, 11622434, 11974382, 11604369, 11974080, 11704398, and 11874090), the
National Key Research and Development Program of China under Grant No.
2017YFA0304402, the CAS Youth Innovation Promotion Association (Grant Nos.
Y201963 and 2018364), the Hubei Province Science Fund for Distinguished Young
Scholars (Grant No. 2017CFA040), and the K. C. Wong Education Foundation
(Grant No. GJTD-2019-15). ZCY was supported by NSERC of Canada. Work of A.D.
was supported in part by the U.S. National Science Foundation.

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