# Precise determination of ground-state hyperfine splitting and calculation of
Zeeman coefficients for 171Yb+ microwave frequency standard

J. Z. Han  State Key Laboratory of Precision Measurement Technology and
Instruments, Key Laboratory of Photon Measurement and Control Technology of
Ministry of Education, Department of Precision Instrument, Tsinghua
University, Beijing 100084, China  B. Q. Lu  National Time Service Center,
Chinese Academy of Sciences, Xi’an 710600, China  N. C. Xin  State Key
Laboratory of Precision Measurement Technology and Instruments, Key Laboratory
of Photon Measurement and Control Technology of Ministry of Education,
Department of Precision Instrument, Tsinghua University, Beijing 100084, China
Y. M. Yu<EMAIL_ADDRESS>Beijing National Laboratory for Condensed
Matter Physics, Institute of Physics, Chinese Academy of Sciences, Beijing
100190, China  University of Chinese Academy of Sciences, Beijing 100049,
China  H. R. Qin  Department of Physics, Tsinghua University, Beijing 100084,
China  S. T. Chen  State Key Laboratory of Precision Measurement Technology
and Instruments, Key Laboratory of Photon Measurement and Control Technology
of Ministry of Education, Department of Precision Instrument, Tsinghua
University, Beijing 100084, China  Y. Zheng  Department of Physics, Tsinghua
University, Beijing 100084, China  J. G. Li<EMAIL_ADDRESS>Institute
of Applied Physics and Computational Mathematics, Beijing 100088, China  J. W.
Zhang<EMAIL_ADDRESS>State Key Laboratory of Precision Measurement
Technology and Instruments, Key Laboratory of Photon Measurement and Control
Technology of Ministry of Education, Department of Precision Instrument,
Tsinghua University, Beijing 100084, China  L. J. Wang
<EMAIL_ADDRESS>State Key Laboratory of Precision Measurement
Technology and Instruments, Key Laboratory of Photon Measurement and Control
Technology of Ministry of Education, Department of Precision Instrument,
Tsinghua University, Beijing 100084, China  Department of Physics, Tsinghua
University, Beijing 100084, China

###### Abstract

We report precise measurement of the hyperfine splitting and calculation of
the Zeeman coefficients of the 171Yb+ ground state. The absolute hyperfine
splitting frequency is measured using high-resolution laser-microwave double-
resonance spectroscopy at 0.1 mHz level, and evaluated using more accurate
Zeeman coefficients. These Zeeman coefficients are derived using Landé $g_{J}$
factors calculated by two atomic-structure methods, multiconfiguration Dirac-
Hartree-Fock, and multireference configuration interaction. The cross-check of
the two calculations ensures an accuracy of the Zeeman coefficients at
$10^{-2}$ Hz/G2 level. The results provided in this paper improve the accuracy
and reliability of the second-order Zeeman shift correction, thus further
improving the accuracy of the microwave frequency standards based on 171Yb+.
The high-precision hyperfine splitting and Zeeman coefficients could also
support could also support further experiments to improve the constraints of
fundamental constants through clock frequency comparison of the Yb+ system.

††preprint: APS/123-QED

##  I Introduction

The microwave frequency standard is one of the most widely used quantum
technologies. Those based on trapped ions are considered the next generation
of atomic frequency standards due to their potentially high stability, high
accuracy, and transportability Schmittberger and Scherer (2020). Such
frequency standards are expected to promote development in areas such as
satellite navigation Bandi  _et al._ (2011); Mallette  _et al._ (2010), deep
space exploration Prestage and Weaver (2007); Liu  _et al._ (2018); Burt  _et
al._ (2021), timekeeping Diddams  _et al._ (2004); Burt  _et al._ (2008) and
telecommunications Ho  _et al._ (1998). At present, trapped-ion microwave
frequency standards are mostly based on 199Hg+ Berkeland  _et al._ (1998);
Burt  _et al._ (2021), 113Cd+ Tanaka  _et al._ (1996); Jelenković  _et al._
(2006); Qin  _et al._ (2022) and 171Yb+ Phoonthong  _et al._ (2014);
Mulholland  _et al._ (2019a); Xin  _et al._ (2022). The clock transition of
those candidates is their ground-state hyperfine splittings.

171Yb+ ion, which benefits from a simple structure ($I=1/2$) and relatively
large ground-state hyperfine splitting ($\nu_{\rm HFS}=12.6$-GHz), is very
suitable for the development of microwave frequency standards. More
importantly, lasers for cooling and repumping 171Yb+ ions can be easily
obtained by compact semiconductor lasers and are fiber-friendly. The above
advantages make the laser-cooled 171Yb+ scheme stand out among other
candidates for trapped-ion microwave frequency standards for practical use. In
addition, the 171Yb+ ion is also widely used in areas such as optical
frequency standards Schneider  _et al._ (2005); Huntemann  _et al._ (2016),
quantum computation Lu  _et al._ (2019); Morong  _et al._ (2021); Yang  _et
al._ (2022), and searching for new physics based on trapped ions Godun  _et
al._ (2014); Huntemann  _et al._ (2014); Counts  _et al._ (2020); Lange  _et
al._ (2021). Several metrology laboratories have focused on developing
microwave frequency standards based on trapped 171Yb+ ions. For instance, the
National Institute of Information and Communications in Japan has developed a
laser-cooled 171Yb+ ion microwave frequency standard with a stability of
$2.1\times 10^{-12}/\sqrt{\tau}$ and an evaluated accuracy of $1.1\times
10^{-14}$ Phoonthong  _et al._ (2014). The National Physical Laboratory in the
UK has built a prototype with a stability of $3.6\times 10^{-12}/\sqrt{\tau}$.
The entire system fits into a 6U 19-inch rack unit ($51\times 49\times 28$)
cm3, revealing its potential for miniaturization Mulholland  _et al._ (2019b,
a). The Sandia National Laboratories, the Jet Propulsion Laboratory, and the
National Institutes of Standards and Technology in the USA have developed a
chip-scaled 171Yb+ microwave frequency standard. The vacuum system is several
cm3 in size and achieved a stability of $2\times 10^{-11}/\sqrt{\tau}$
Schwindt  _et al._ (2015, 2016). Microchip Technology in the USA has developed
a buffer gas-cooled 171Yb+ microwave frequency standard for military and
commercial applications Park  _et al._ (2020). The National Measurement
Laboratory in Australia has also predicted a frequency stability of better
than $5\times 10^{-14}$ and a frequency uncertainty of $4\times 10^{-15}$ for
a laser-cooled 171Yb+ microwave frequency standard Park  _et al._ (2007).

Our group at Tsinghua University has been committed to developing microwave
frequency standards based on laser-cooled Cd+ Zhang  _et al._ (2012); Wang
_et al._ (2013); Miao  _et al._ (2015); Zuo  _et al._ (2019); Han  _et al._
(2019a, b, 2021); Miao  _et al._ (2021); Han  _et al._ (2022a, b); Qin  _et
al._ (2022) and Yb+ Xin  _et al._ (2022) for the past fifteen years. In
particular, we have achieved the most stable laser-cooled 171Yb+ microwave
frequency standard ($8.5\times 10^{-13}/\sqrt{\tau}$) and aim to evaluate its
accuracy to the $10^{-15}$ level Xin  _et al._ (2022). A high accuracy,
transportable 171Yb+ microwave frequency standard can provide a better time-
frequency reference for the China’s BeiDou Navigation Satellite System (BNSS).
However, to achieve such a high accuracy, all frequency shifts of the clock
transition need to be measured or evaluated down to 0.1 mHz level accuracy.
These pose a challenge to both experimentalists and theorists.

Accurate frequency comparison of different clock transitions between atomic
levels of the electronic or hyperfine transitions over time also can be used
to search for new boson Counts  _et al._ (2020), to test local position
invariance Lange  _et al._ (2021), and to investigate variations of the fine
structure constant $\alpha$ Godun  _et al._ (2014) and the proton-to-electron
mass ratio $\mu$ Huntemann  _et al._ (2014). For instance, constraints of
$\dot{\alpha}/\alpha<-0.20(20)\times 10^{-16}$/yr and
$\dot{\mu}/\mu<-0.5(1.6)\times 10^{-16}$/yr are given using frequency
comparisons of the 171Yb+ 467-nm E3 optical transition and 133Cs 9.2-GHz
ground-state hyperfine splitting Huntemann  _et al._ (2014). The constraints
may be further improved by using frequency comparison of the 171Yb+ 467-nm
optical clock transition and 171Yb+ 12.6-GHz ground-state hyperfine splitting,
as some common-mode noise can be suppressed. Such frequency comparison also
relies on accurately evaluating the absolute clock frequencies, especially the
12.6-GHz ground-state hyperfine splitting frequency in 171Yb+.

Among all the systematical energy shifts of the measurement of the the clock
frequency, i.e., the ground-state hyperfine splitting, the second-order Zeeman
shift caused by the magnetic field is dominant Berkeland  _et al._ (1998);
Phoonthong  _et al._ (2014); Qin  _et al._ (2022). Note that the second-order
Zeeman shift of in a trapped-ion microwave frequency standard is approximately
four orders of magnitude larger than other shifts, even in a strong magnetic
shielding environment composed of five layers of $\mu$-mental and one layer of
soft iron Miao  _et al._ (2021); Qin  _et al._ (2022). Therefore, the second-
order Zeeman shift must be carefully evaluated. For the
$6s~{}^{2}S_{1/2}~{}(F=0)\rightarrow(F=1)$ 12.6-GHz hyperfine splitting, the
second-order Zeeman shift $\Delta\nu_{\rm SOZS}$ can be expressed as Itano
(2000),

$\Delta\nu_{\rm
SOZS}=K_{0}B_{0}^{2}=\frac{(g_{J}-g^{\prime}_{I})^{2}\mu_{B}^{2}}{2h^{2}\nu_{\rm
HFS}}B_{0}^{2},$ (1)

where $K_{0}=(g_{J}-g^{\prime}_{I})^{2}\mu_{B}^{2}/(2h^{2}\nu_{\rm HFS})$ is
the second-order Zeeman coefficient, $\nu_{\rm HFS}$ is the ground-state
hyperfine splitting, $g_{J}$ and $g^{\prime}_{I}$ are the electronic and
nuclear $g$ factor, $h$ is the Planck constant and $\mu_{B}$ is the Bohr
magneton. Therefore, the accuracy of $K_{0}$ is mainly limited by the accuracy
of $g_{J}$ since $\nu_{\rm HFS}$ and $g^{\prime}_{I}$ are accurate enough.
However, there are only two theoretical results and one early measurement of
the ground-state Landé $g_{J}$ factor in 171Yb+ and the results are not
consistent: one giving 2.002798(113), calculated by relativistic-coupled-
cluster (RCC) theory Yu  _et al._ (2020); another giving 2.003117 without
specifying uncertainty, calculated by time-dependent Hartree-Fock (TDHF)
method Gossel  _et al._ (2013); and the other giving 1.998 without specifying
uncertainty, an early spectroscopic measurement Meggers (1967). Even for two
theoretical results, there exists a difference of around 0.0003 that generates
a fractional second-order Zeeman shift of about $(0.8\sim 7.9)\times 10^{-14}$
in a typical static magnetic field ($0.03\sim 0.10$ G) Berkeland  _et al._
(1998); Phoonthong  _et al._ (2014); Qin  _et al._ (2022) for operating
trapped-ion microwave frequency standard. The large uncertainty of second-
order Zeeman shift caused by the larger error bar of the Landé $g_{J}$ factor
can obviously deteriorate the accuracy of the latest 171Yb+ microwave
frequency standard of our group (goal $<9\times 10^{-15}$ Xin  _et al._
(2022)) and others ($1.1\times 10^{-14}$ Phoonthong  _et al._ (2014)).

In this work, we report on the precise determination of the ground-state
hyperfine splitting and calculation of Zeeman coefficients for 171Yb+
microwave frequency standard. The hyperfine splitting is measured using laser-
microwave double-resonance spectroscopy Vogel and Vogel (2018) in our laser-
cooled 171Yb+ microwave frequency standard and evaluated using the new Zeeman
coefficients obtained in this paper. Those coefficients are derived using the
Landé $g_{J}$ factor calculated by two atomic structure calculation methods in
this paper: the multiconfiguration Dirac-Hartree-Fock (MCDHF) and the
multireference configuration interaction (MRCI). The cross-check of the new
Zeeman coefficients guarantee clock transition evaluation of $10^{-15}$
accuracy under typical conditions. The results reported in this paper are of
great importance for further improving the performance of the microwave
frequency standards based on trapped 171Yb+ ions. Our efforts on the ground
state Landé $g_{J}$ factor and hyperfine splitting could also support further
experiments in improving the constraints of fundamental constants through
clock frequency comparison of the Yb+ system.

##  II Measurement of hyperfine splitting $\nu_{\rm HFS}$

Figure 1: (a) Schematic of the experimental setup for measuring the hyperfine
splitting $\nu_{\rm HFS}$ of 171Yb+. Lasers with wavelengths of 369 nm, 399
nm, and 935 nm are used for cooling & probing, ionizing, and repumping of
171Yb+. $\nu_{\rm HFS}$ is measured under a closed-loop operation. PMT,
photomultiplier tubes; OCXO, oven-controlled crystal oscillator; PID,
proportional-integral-differentiation. (b) Schematic energy levels of the
171Yb+ ion and 171Yb atom (not to scale).

Figure 2: (a) Typical Ramsey fringe of the clock transition (12.6-GHz) with a
free evolution time of 1000 ms, a $\pi/2$ microwave pulse time of 60 ms, a
microwave power of -29.8 dBm, and a fluorescence signal integration time of
150 ms. The central frequency and linewidth of the Ramsey fringe are
12642812121.47 Hz and 434 mHz. (b) Rabi fringe of the Zeeman transition with a
$\pi$ microwave pulse time of 60 ms, microwave power of -3 dBm, and scan step
of 400 Hz. Peaks from left to right are $F=1,m_{F}=-1\rightarrow F=0,m_{F}=0$
and $F=1,m_{F}=1\rightarrow F=0,m_{F}=0$ transitions, respectively.

The hyperfine splitting $\nu_{\rm HFS}$ is measured through a laser-microwave
double-resonance spectroscopy in our laser-cooled 171Yb+ microwave frequency
standard. The experiment is conducted in a linear Paul radio-frequency (RF)
ion trap, and the schematic of the setup is shown in Fig. 1 (a). The ratio of
$R=r_{0}$ is optimized and set at 1.1468, for which $2R=14.22$ mm is the outer
diameter of the electrode and $r0=6.2$ mm is the radial distance from the axis
of the trap to the closest surface of the electrodes, to reduce heating
effects from higher-order RF potential and to increase the number of trapped
ions. A magnetic shield of three layers of $\mu$-metal is installed outside
the vacuum chamber to shield the external magnetic field. Approximately
$10^{5}$ 171Yb+ ions are loaded in the ion trap for hyperfine splitting
measurement, and the energy levels of 171Yb+ are shown in Fig. 1 (b). The ions
are photon-ionized from 171Yb atoms by a 399-nm (Toptica DLpro,
$6s^{2}~{}^{1}S_{0}\rightarrow 6s6p~{}^{1}P_{1}$) and a 369-nm (Precilasers
FL-SF, $6s6p~{}^{1}P_{1}\rightarrow$ ionized continuum) laser beam. The
temperature of ions is reduced from laser Doppler cooling by a 369-nm
($6s~{}^{2}S_{1/2}\rightarrow 6p~{}^{2}P_{1/2}$) laser beam to reduce the
second-order Doppler shift. A 935-nm laser beam (Toptica DLpro,
$6d~{}^{2}D_{3/2}\rightarrow 5d6s~{}^{3}[3/2]_{1/2}$) and 12.6-GHz microwave
radiation (Agilent E8257D, $6s~{}^{2}S_{1/2}~{}F=0\rightarrow
6s~{}^{2}S_{1/2}~{}F=1$) are applied to repump the ions from ${}^{2}D_{3/2}$
and ${}^{2}S_{1/2}~{}(F=0)$ dark states back to the cooling cycle. The
frequency of each laser beam is measured and stabilized using a high-precision
wavemeter (HighFinesse WS8-2). Three pairs of Helmholtz coils installed near
the vacuum chamber are used to provide the quantization axis (several
$10^{-2}$ G) of the 171Yb+ ions and compensate the geomagnetic field. Another
pair of Helmholtz coils generate a 7-G strong magnetic field. The direction of
the strong magnetic field magnetic field forms an angle of approximately
$30^{\circ}$ with the 369-nm laser beam to induce a precession in the dipole
moments of the ions and to destabilize the Zeeman dark states of the
${}^{2}S_{1/2}~{}(F=1)$ level Berkeland and Boshier (2002).

After the ions are loaded and cooled, the hyperfine splitting $\nu_{\rm HFS}$
is measured by Ramsey’s method of separated oscillatory fields in a closed-
loop operation. Large ion cloud in a linear ion trap suffers from RF heating,
unlike single ion or ion string. The optimized ion trap ensures that up to
$10^{5}$ 171Yb+ can be stably trapped in the ion trap at a relatively low
temperature during the clock signal’s interrogation. In the passive quantum
frequency standard, the signal-to-noise ratio (SNR) is proportional to
$\sqrt{N}$, where $N$ is the number of quantum particles. Thus, $10^{5}$ of
ions ensures their clock transition can be detected with a high SNR. A
typical, high SNR Ramsey fringe of the clock transition detected by
photomultiplier tubes (PMT, Hamamatsu H12386-210) is shown in FIG. 2 (a),
where the full width at half maximum (FWHM) of the center fringe is
$2\delta=434$ mHz. The output frequencies of the oven-controlled crystal
oscillator (OCXO, Rakon HSO14) are locked by the central peak of the Ramsey
fringe through a proportional-integral-differentiation (PID) controller and
are recorded by comparison with the H-maser. The clock transition under a
static magnetic field $\nu_{\rm HFS}(B_{0})$ can be calculated by using

$\displaystyle\nu_{\rm clock}(B_{0})$ $\displaystyle=$ $\displaystyle f_{\rm
OCXO}\times M_{\rm Synthesizer}$ (2)

where $f_{\rm OCXO}$ is the average value of the output frequency of the OCXO,
$M_{\rm Synthesizer}$ is the magnification of the microwave synthesizer.

The static magnetic field $B_{0}$ experienced by the ions can be calibrated by
measuring the Larmor frequency difference $\Delta\nu_{L}$ of the
$6s~{}^{2}S_{1/2}~{}(F=0,m_{F}=0)\rightarrow(F=1,m_{F}=\pm 1)$ Zeeman
sublevels. The Larmor frequency is measured by using the Rabi oscillation
spectra, as shown in Fig. 2 (b). The static magnetic field $B$ can be
calculated by using

$B_{0}=\frac{\Delta\nu_{L}}{(g_{J}+g^{\prime}_{I})\mu_{B}/h}=\frac{\Delta\nu_{L}}{2K_{Z}},$
(3)

where $K_{Z}=(g_{J}+g^{\prime}_{I})\mu_{B}/(2h)$ is the first-order Zeeman
coefficient. The second-order Zeeman shift of the hyperfine splitting can be
evaluated by using Eq. (1). The absolute hyperfine splitting $\nu_{\rm HFS}$
can be derived by $\nu_{\rm HFS}(B_{0})$ subtracting the second-order Zeeman
shift and other systematic frequency shifts.

##  III THEORY

###  III.1 Operator of Landé $g_{J}$ factor

The interaction Hamiltonian between an atom and the magnetic field can be
written as Andersson and Jönsson (2008)

$\displaystyle H_{m}=(\boldsymbol{N^{(1)}}+\boldsymbol{\Delta N^{(1)}})B,$ (4)

by choosing the direction of the magnetic field as the $z$-direction and
neglecting all diamagnetic contributions in a relativistic frame. The
electronic tensor operators of an N-electron atom can be expressed as Cheng
and Childs (1985),

$\displaystyle\boldsymbol{N^{(1)}}$ $\displaystyle=$
$\displaystyle\sum^{N}_{i=1}\boldsymbol{n^{(1)}}(i)=\sum^{N}_{i=1}-I\frac{\sqrt{2}}{2\alpha}r_{i}(\boldsymbol{\alpha_{i}}\boldsymbol{C^{(1)}}(i))^{(1)},$
$\displaystyle\boldsymbol{\Delta N^{(1)}}$ $\displaystyle=$
$\displaystyle\sum^{N}_{i=1}\boldsymbol{\Delta
n^{(1)}}(i)=\sum^{N}_{i=1}\frac{g_{s}-2}{2}\beta_{i}\boldsymbol{\Sigma_{i}},$
(5)

where $\Sigma_{i}$ is the relativistic spin-matrix, $I$ is the imaginary unit,
and $g_{s}=2.00232$ is the g factor of the electron spin corrected for QED
effects. The $\boldsymbol{\Delta N^{(1)}}$ term is the Schwinger QED
correction. The interaction Hamiltonian $H_{m}$ can be treated in first-order
perturbation theory in a weak magnetic field situation. A fine-structure level
$\Gamma J$ is split according to

$\displaystyle\langle\Gamma JM_{J}|N_{0}^{(1)}+\Delta N_{0}^{(1)}|\Gamma
JM_{J}\rangle B$ (6) $\displaystyle=$
$\displaystyle\frac{M_{J}}{\sqrt{J(J+1)}}\langle\Gamma J||N_{0}^{(1)}+\Delta
N_{0}^{(1)}||\Gamma J\rangle B$ $\displaystyle=$ $\displaystyle
g_{J}M_{J}\frac{B}{2},$

where $J$ is total angular momentum, $M_{J}$ is component along the $z$
direction of $J$, $\Gamma$ represent other appropriate angular momentum, and
Landé $g_{J}$ is defined as

$g_{J}=2\frac{\langle\Gamma J||N_{0}^{(1)}+\Delta N_{0}^{(1)}||\Gamma
J\rangle}{\sqrt{J(J+1)}}.$ (7)

###  III.2 Multiconfiguration Dirac-Hartree-Fock approach

The MCDHF method Jönsson  _et al._ (2022), as implemented in the Grasp package
Jönsson  _et al._ (2023), is employed to obtain wave functions referred to as
the atomic state function (ASF). The ASF are approximate eigenfunctions of the
Dirac Hamiltonian describing a Coulombic system,

$\displaystyle H_{\rm DC}$ $\displaystyle=$
$\displaystyle\sum_{i}[c(\vec{\bm{\alpha}}\cdot\vec{\bf
p})_{i}+(\bm{\beta}_{i}-1)c^{2}+V_{nuc}(r_{i})]+\sum_{i<j}^{N}\frac{1}{r_{ij}}$
(8)
$\displaystyle-\frac{1}{2r_{ij}}[\vec{\bm{\alpha_{i}}}\cdot\vec{\bm{\alpha_{j}}}+\frac{(\vec{\bm{\alpha_{i}}}\cdot\vec{\bm{r_{ij}}})(\vec{\bm{\alpha_{j}}}\cdot\vec{\bm{r_{ij}}})}{r_{ij}^{2}}],$

where $\vec{\bm{\alpha}}$ and $\bm{\beta}$ represent the Dirac matrices,
$\vec{\bf p}$ is the momentum operator, $r_{ij}$ is the distance between
electrons $i$ and $j$, and $V_{nuc}(r)$ is the nuclear potential results from
a nuclear charge density given by a two-parameter Fermi distribution function
Parpia and Mohanty (1992). The last term is the Breit interaction in the low-
frequency approximation. The ASF is a linear combination of configuration
state functions Grant (2007),

$\displaystyle\Psi(\Gamma JM_{J})=\sum^{N_{\rm
CSF}}_{i=1}c_{i}\Phi(\Gamma_{i}JM_{J}),$ (9)

where $c_{i}$ represents the mixing coefficient corresponding to the $i$th
configuration state function. The CSFs are the linear combinations of one-
electron Dirac orbital products.

The active space approach is adopted to capture electron correlations in this
work. Our computational model is shown in FIG. 3. The computation started from
the Dirac-Hartree-Fock (DHF) approximation. The occupied spectroscopic
orbitals are optimized in the single reference configuration $\\{[\rm
Ne]3s^{2}3p^{6}3d^{10}4s^{2}4p^{6}4d^{10}4f^{14}5s^{2}5p^{6}~{}6s\\}$.
According to perturbation theory, electron correlations can be divided into
first-order and higher-order correlations. The outermost $6s$ orbitals in the
reference configuration are treated as the valence electrons, and the others
are treated as the core. The correlations between the valence and core
electrons (CV) are considered in the self-consistent field (SCF) calculations.
The CV correlations are taken into account using the single and restricted
double (SrD) substitutions from the $n\geq 3$ core orbitals. The SrD
substitutions mean that only one electron in each core orbital can be excited
to the active set. The active set is expanded layer by layer to
$\\{14s,13p,12d,12f,10g,10h,8i\\}$, and only the added correlation orbitals
are optimized each time. The correlations between core electrons (CC) are
considered in the RCI calculations where only the mixing coefficients are
variable. The CC correlations are accounted by CSFs generated through SD
subsitituions from allowing the single and double (SD) substitutions from the
$4f,5s,5p$ core orbitals to the largest active set. The main higher-order
correlations are captured by the multireference (MR) SD-excitation method Li
_et al._ (2012); Filippin  _et al._ (2016); Bieroń  _et al._ (2009). The MR
configuration set was formed by selecting the dominant CSFs in the CC model,
i.e., those CSFs with mixing coefficients $c_{i}$ larger than 0.02. Finally,
the Breit interaction and QED (vacuum polarization and self-energy) are
considered in the RCI procedure. CSFs that do not interact with the reference
configurations are removed to raise the computational efficiency Jönsson  _et
al._ (2007); Froese-Fischer  _et al._ (2022).

Figure 3: Computational model of the MCDHF method. The model takes into
account the main first-order and higher-order electron correlations. The
first-order electron correlations include the CV and CC correlations.
Corrections for the Breit and QED effects are added in the final result.

###  III.3 Multireference Configuration Interaction approach

The calculation of the Yb+ ion is carried out by using the multireference
configuration interaction (MRCI) method that is based on the general active
space (GAS) Fleig  _et al._ (2001, 2003, 2006) and implemented in the KR-CI
module of the Dirac package Knecht  _et al._ (2008); Jensen  _et al._ (2022).
We start with the close-shell Dirac-Hartree-Fock (DHF) calculation of Yb2+
under the Dirac-Coulomb-Gaunt Hamiltonian. The Dirac-Coulomb-Gaunt Hamiltonian
is written as

$\displaystyle H_{\rm DCG}$ $\displaystyle=$
$\displaystyle\sum_{i}[c(\vec{\bm{\alpha}}\cdot\vec{\bf
p})_{i}+(\bm{\beta}_{i}-1)c^{2}+V_{nuc}(r_{i})]+$ (10)
$\displaystyle\sum_{i<j}\bigg{[}\frac{1}{r_{ij}}-\frac{{\vec{\bm{\alpha}}}_{i}\cdot{\vec{\bm{\alpha}}}_{j}}{2r_{ij}}\bigg{]}.$

The last term in Eq. (10) represents the Gaunt interaction, which is the
leading term of the Breit interaction.

In the DHF calculation, a Gaussian basis set of quadruple-zeta quality,
dyall.cv4z, is adopted Gomes  _et al._ (2010). Next, the electron correlation
calculation is performed based on the optimized spinors that are obtained by
the DHF calculation. We correlate the outermost 23 electrons of Yb+, which
takes into account the single (S) excitation from the $5s$ and $5p$ shells,
the single and double (SD) excitation from the $4f$ shell and the single,
double and triple (SDT) excitation from the $6s$ and $5d$ valent shells. The
virtual energy orbitals with energy above 10 a.u. are truncated off from the
configuration interaction calculation, considering their negligible impact on
the MRCI results. The DHF and MRCI calculations are carried out under the
Dirac package.

##  IV Results & Discussion

Table 1: Landé $g_{J}$ factor, first-order Zeeman coefficient ($K_{Z}$, in
$10^{3}$ Hz/G), second-order Zeeman coefficient ($K_{0}$, in Hz/G2) and
hyperfine splitting ($\nu_{\rm HFS}$, in mHz) of ground-state 171Yb+ obtained
in this work. The $K_{0}$ coefficients are calculated based on $\nu_{\rm
HFS}=12642812118.47(1)$ Hz. Results of other works are also listed for
comparison. Landé $g_{J}$ factor
---
$g_{J}$ | Source |
2.002630(38) | MCDHF (This paper) | Theor.
2.002604(55) | MRCI (This paper) | Theor.
2.002617(68) | Final (This paper) | Theor.
2.002798(113) | RCC Yu  _et al._ (2020) | Theor.
2.003117 | TDHF Gossel  _et al._ (2013) | Theor.
2.0023 | Nonrelat. Vanier and Audoin (1989) | Theor.
1.998 | Spectr. Meggers (1967) | Expt.
Zeeman coefficients
$K_{0}$ | $K_{Z}$ | Source |
310.874(12) | 1401.089(27) | MCDHF (This paper) | Theor.
310.866(17) | 1401.071(38) | MRCI (This paper) | Theor.
310.870(21) | 1401.080(47) | Final (This paper) | Theor.
310.77 | 1400.86 | Nonrelat. Vanier and Audoin (1989) | Theor.
310.93(35) | 1401.206(79) | RCC Yu  _et al._ (2020)a | Theor.
311.03 | 1401.430 | TDHF Gossel  _et al._ (2013)a | Theor.
309a | 1398a | Spectr. Meggers (1967)a | Expt.
Hyperfine splitting
$\nu_{\rm HFS}$ | Source |
12642812118468.9(8) | This paper | Expt.
12642812118468.2(4) | Phoonthong  _et al._ (2014) | Expt.
12642812118468.5(9) | Warrington  _et al._ (2002) | Expt.
12642812118466(2) | Fisk  _et al._ (1995) | Expt.
12642812118471(9) | Tamm  _et al._ (1995) | Expt.
12642812118468(16) | Sellars  _et al._ (1995) | Expt.
aDerived by using the Landé $g_{J}$ factors given in the Refs.

###  IV.1 Landé $g_{J}$ factor & Zeeman coefficients

The Landé $g_{J}$ factor, the first- and second-order Zeeman coefficient, and
the hyperfine splitting of the 171Yb+ ground-state obtained in this work are
shown in TABLE 1. The Landé $g_{J}$ factors calculated by the MCDHF and MRCI
methods are in excellent agreement, and the deviation between the two is at
the fifth decimal place. Uncertainties of Landé $g_{J}$ factors are given in
parentheses. Accurate calculations of Landé $g_{J}$ factors have proven to be
complicated even for alkali-metal atoms and alkali-metal-like ions Sahoo and
Kumar (2017), as they are sensitive to electron correlations effects. Such
effects of the electrons from the $4f$ orbitals and higher-order corrections
in Yb+ are also proved to be large and complicated. The inconsistency between
the theory and experiment of the quadruple moment in Yb+ $4f~{}^{2}F_{7/2}$
state is also due to the above reason Nandy and Sahoo (2014). These two
results are within the uncertainty of the previous RCC result Yu  _et al._
(2020) and improve the accuracy of the Landé $g_{J}$ factor from the fourth to
the fifth decimal place. All three results have a deviation of approximately
0.0004 from the previous results calculated by the TDHF method Gossel  _et
al._ (2013). The theoretical results have a deviation up to 0.005 compared
with the early spectroscopic result given by Meggers Meggers (1967). The
anomaly correction of the Landé $g_{J}$ factor has the opposite sign compared
with the theoretical results. A recent measurement of the second-order Zeeman
coefficient of the ${}^{2}D_{5/2}$ in 171Yb+ also notwithstanding with the
early spectroscopic result Tan  _et al._ (2021). Those suggests potential
issues in the early spectroscopic result of the Landé $g_{J}$ factor in Yb+.
Comparing the results of the same physical constant from different calculation
methods is also crucial for investigating the role of electronic correlations
and for developing atomic structure calculation theory. Therefore, we
encourage more experimental and theoretical research on the Landé $g_{J}$
factors in Yb+.

The $K_{0}$ and $K_{Z}$ coefficients of 171Yb+ ground-state are calculated
using Eq. (1) and Eq. (3), where $g_{J}=2.002617(68)$ is the Landé $g_{J}$
factor recommended in this paper, $\nu_{\rm HFS}=12642812118.47(1)$ Hz is the
hyperfine splitting, and $g^{\prime}_{I}=-\mu_{\rm
Yb}\cdot\mu_{N}/(I\cdot\mu_{B})=-0.53772(1)\times 10^{-3}$ is the nuclear
g-factor of 171Yb+ Olschewski (1972); Tiesinga  _et al._ (2021). The
uncertainties of those coefficients are given in parentheses. The widely used
values of $K_{0}$ and $K_{Z}$ are $310.77$ Hz/${\rm G}^{2}$ and $1400.86\times
10^{3}$ Hz/${\rm G}$ standards without specifying uncertainty for quantum
frequency standard community Vanier and Audoin (1989). Those coefficients are
calculated using a non-relativistic Landé $g_{J}$ factor of 2.0023. As seen
from the previous analysis, those coefficients cannot meet the requirements of
the state-of-the-art 171Yb+ microwave frequency standards. The accuracy of the
clock transition may be underestimated without considering the uncertainty of
the Zeeman coefficients. The more accurate and reliable $K_{0}$ and $K_{Z}$
coefficients calculated in this work guarantee the evaluation accuracy of the
171Yb+ microwave frequency standard. Zeeman coefficients calculated using
Landé $g_{J}$ factor of other works are also listed in TABLE 1 for comparison.

###  IV.2 hyperfine splitting $\nu_{\rm HFS}$

The ground-state hyperfine splitting $\nu_{\rm HFS}$ is determined in our high
stability ($8.5\times 10^{-13}/\sqrt{\tau}$) laser-cooled 171Yb+ microwave
frequency standard. The clock transition under a static magnetic field
$\nu_{\rm HFS}(B_{0})$ is calculated to be 12642812121757.4(4) mHz by using
Eq. (2) where $f_{\rm OCXO}$ is 9999999083.85136 mHz and $M_{\rm Synthesizer}$
is 1264.2813280027. The value shown in parenthesis is the statistical
uncertainty estimated from Allan deviation data at 1000 s Xin  _et al._
(2022). The Larmor frequency difference of the Zeeman transition
$\Delta\nu_{L}$ is measured to be 288198(29) Hz. The static magnetic field
$B_{0}$ is then calibrated to be 0.102849(10) G using Eq. (3) with the
measured $\Delta\nu_{L}$ and the calculated $K_{Z}$. Therefore, the second-
order Zeeman shift $\Delta\nu_{\rm SOZS}$ is evaluated to be 3288.32(67) mHz
using Eq. (1) with the calculated $K_{0}$ in this paper. In addition, the
reference shift caused by the frequency difference between the H-maser and the
primary frequency standards is measured to be 0.555(6) mHz by the method of
GPS common-view. The black-body radiation Zeeman shift (BBRZS) is estimated to
be $-1.2\times 10^{-4}$ mHz using the method in Ref. Han  _et al._ (2019a).
Frequency shifts, including the second-order Doppler shift (SODS), the black-
body radiation Stark shift (BBRSS), the quadratic Stark shift (QSS), and the
gravitational Redshift (GRS) are estimated to be -0.517(40) mHz (see Ref. Xin
_et al._ (2022) for detail). Estimated systematic shifts and uncertainties are
shown in TABLE 2.

Table 2: Magnitudes (in mHz), uncertainties (Uncert., in mHz) and Fractional uncertainties (Frac., in $10^{-14}$) of the estimated systematic frequency shifts, hyperfine splitting under static magnetic field ($\nu_{\rm HFS}(B_{0})$) and the estimated absolute hyperfine splitting ($\nu_{\rm HFS}$) in 171Yb+. The uncertainty of $\nu_{\rm HFS}$ is the square root of the sum of the square of the uncertainty in $\nu_{\rm HFS}(B_{0})$ and the uncertainty in total systematic shift. The uncertainties of the SOZS stem from $B_{0}$ and $K_{0}$ are listed separately for comparison. The predicted uncertainties of the SOZS in the improved system are also listed. Item | Magnitude | Uncert. | Frac. | Source
---|---|---|---|---
SOZS (Current) | 3288.32 | 0.64 | 5.06 | $B_{0}$
($0.1\pm 10^{-5}$ G) |  | 0.22 | 1.74 | $K_{0}$
SOZS (Improved) |  | 0.02 | 0.15 | $B_{0}$
($0.03\pm 10^{-6}$ G) |  | 0.02 | 0.15 | $K_{0}$
Reference | 0.56 | 0.01 | 0.05 |
SODS | -0.44 | 0.04 | 0.30 |
GRS | 0.06 | 0.00 | 0.00 |
QSS | -0.01 | 0.00 | 0.00 |
BBRSS | -0.13 | 0.01 | 0.10 |
BBRZS | 0.0 | 0.00 | 0.00 |
Pressure | 0.0 | 0.00 | 0.00 |
Light | 0.0 | 0.00 | 0.00 |
Total | 3288.36 | 0.70 | 5.50 |
$\nu_{\rm HFS}(B_{0})$ | 12642812121757.4 | 0.4 | 3.0 |
$\nu_{\rm HFS}$ | 12642812118468.9 | 0.8 | 6.3 |

Thus, the absolute frequency of the ground-state hyperfine splitting $\nu_{\rm
HFS}$ of 171Yb+ is determined to be 12642812118468.9(8) mHz by subtracting
systematic frequency shifts, as listed in TABLE 2. The uncertainty of
$\nu_{\rm HFS}$ is the square root of the sum of the square of the uncertainty
in $\nu_{\rm HFS}(B_{0})$ and the uncertainty in total systematic shift. The
measurement accuracy of $\nu_{\rm HFS}$ is mainly limited by the second-order
Zeeman shift. The fractional uncertainties of the $\Delta\nu_{\rm SOZS}$
caused by the $B_{0}$ and $K_{0}$ are $5.06\times 10^{-14}$ and $1.74\times
10^{-14}$ respectively. The large fractional uncertainties mainly stem from
the large value of the static magnetic field ($B_{0}\approx 0.1$ G) in our
experiment and the fluctuation of the external magnetic field ($\approx
10^{-5}$ G). The large $B_{0}$ value is due to the magnetization of the vacuum
chamber by the strong magnetic field (7 G). Such magnetization can be
mitigated using a nonmagnetic (e.g. Ti) vacuum chamber. The strong magnetic
field can also be eliminated using a polarization-modulating scheme to remove
hyperfine dark states. The $B_{0}$ value can be reduced to around $0.03$-G
after those improvements Berkeland  _et al._ (1998); Phoonthong  _et al._
(2014). In addition, the fluctuation of the external magnetic field can be
further reduced to about $10^{-6}$ G after using more layers of $\mu$-metal
Han  _et al._ (2021). The fractional uncertainties of the $\Delta\nu_{\rm
SOZS}$ caused by $B_{0}$ and $K_{0}$ can be reduced to about $1.5\times
10^{-15}$ in such a situation. Those improvements are in progress.

The Zeeman coefficients calculated in this paper can guarantee the evaluation
accuracy at the current and the improved situations. The high-precision
$\nu_{\rm HFS}$ constant reported in this work is consistent with previous
results. Our results not only support building a $10^{-15}$-level high-
accuracy 171Yb+ microwave frequency standard in China but also improve the
reliability of $\nu_{\rm HFS}$ in 171Yb+ ground-state and provide a good
benchmark for the atomic structure calculations.

##  V CONCLUSION

We report on the precise determination of ground-state hyperfine splitting and
calculation of Zeeman coefficients for 171Yb+ microwave frequency standard.
The hyperfine splitting is measured using laser-microwave double-resonance
spectroscopy in our laser-cooled 171Yb+ microwave frequency standard. The
first- and second-order Zeeman coefficients are derived using the recommended
Landé $g_{J}$ factor in this paper. The Landé $g_{J}$ factor is calculated
using the MCDHF and MRCI methods. The cross-check of the two methods ensures
the reliability of the calculation results. The second-order Zeeman shift of
$\nu_{\rm HFS}$ is evaluated using the measured Larmor frequency and the new
Zeeman coefficients of this paper. The $\nu_{\rm HFS}$ constant is determined
at the 0.1 mHz level and is consistent with previous results. The calculation
and experiment conducted in this paper are significant for developing
microwave frequency standards but also an excellent addition and benchmark to
the current atomic structure calculations. The calculations and measurements
are critical to building a high-performance, laser-cooled microwave frequency
standard based on trapped 171Yb+ ions in China. These results can also support
further experiments to improve the constraints of fundamental constants
through clock frequency comparison of the Yb+ system.

###### Acknowledgements.

The authors acknowledge R. Si, J. P. Liu, Z. M. Tang, and H. Wang for their
helpful discussions. This work is supported by the National Key R&D Program of
China (No. 2021YFA1402100), the National Natural Science Foundation of China
(No. 12073015), and the Space Application System of China Manned Space
Program. J.Z.H. and B.Q.L. contributed equally to this work.

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