# Be optical lattice clocks with the fractional Stark shift up to the level of
10-19

Lei Wu, Xia Wang, Ting Wang, Jun Jiang†, Chenzhong Dong  Key Laboratory of
Atomic and Molecular Physics and Functional Materials of Gansu Province,
College of Physics and Electronic Engineering, Northwest Normal University,
Lanzhou 730070, China<EMAIL_ADDRESS>
###### Abstract

The energy levels and electric dipole ($E1$) matrix elements of the ground
state and low-lying excited states of Be atoms are calculated using the
relativistic configuration interaction plus core polarization (RCICP) method.
The static and dynamic $E1$, magnetic dipole ($M1$) and electric quadrupole
($E2$) polarizabilities as well as the hyperpolarizabilities of the
$2s^{2}~{}^{1}S_{0}$ and $2s2p~{}^{3}P_{0}$ states are determined. Two magic
wavelengths, 300.03 and 252.28 nm, of $2s^{2}~{}^{1}S_{0}\rightarrow
2s2p~{}^{3}P_{0}$ clock transition are found. Then, the multipolar and
nonlinear Stark shifts of the clock transition at the magic wavelength are
discussed in detail. We find that when the laser intensity $I$ is in the range
of 14.3 $\sim$ 15.9 kW/cm2 and the detuning $\delta$ (the frequency detuning
of the lattice laser frequency relative to the magic frequency) is in the
range of 40.7 $\sim$ 40.9 MHz, the fractional Stark shifts of the clock
transition are less than 1.0 $\times$ 10-18. While, when $I$ is in the range
of 15.01 $\sim$ 15.46 kW/cm2 and $\delta$ is in the range of 40.73 $\sim$
40.76 MHz, the fractional Stark shifts are lower than 1.0 $\times$ 10-19.

Keywords: polarizability, magic wavelengths, multipolar and nonlinear Stark
shifts

##  1 Introduction

In the past few decades, with the rapid development of laser cooling and
trapping techniques, extraordinary advancements in optical atomic clock
accuracy and stability have been demonstrated [1, 2, 3, 4, 5, 6]. The high-
accuracy optical clocks can be used for performing precision measurements of
fundamental physical constants [7, 8], testing the local Lorentz invariance
[9, 10], exploring variations of fine structure constant $\alpha$ with time
[11, 12], probing dark matter and dark energy [13, 14], detecting
gravitational waves [15], and detecting new forces beyond the standard model
of particle physics [15, 14].

Meticulously studying the interaction between laser fields and atoms has
become the core of developing ultra-higher-precision optical clocks. The
interaction between laser fields and atoms can cause ac Stark shift, which
would affect the accuracy of the measurement of relevant atomic parameters. To
reduce the impact of the ac Stark shift, magic-wavelength trapping was
introduced in Refs. [16, 17, 5, 6]. The dynamic electric dipole ($E1$)
polarizabilities of a given pair of energy levels are the same as each other
in the magic-wavelength trapping, and the second-order Stark shift is
eliminated. However, when the accuracy of the optical lattice clock is at the
level of $10^{-18}$, the multipolar (electric quadrupole $E2$ and magnetic
dipole $M1$) and nonlinear Stark shifts are non-negligible [18, 19, 20, 21,
22]. These Stark shifts are related to the $E2$ and $M1$ polarizabilities as
well as hyperpolarizability [18, 23, 20, 19, 21, 24, 25, 26]. For pursuing
much higher precision, the effects on the systematic uncertainty of optical
clocks from the multipolar and nonlinear Stark shifts need to be evaluated
[18, 19, 20, 21, 24, 25].

Be atoms have been proposed as one of the potential candidates for developing
ultra-high-precision optical clocks due to their unique properties [27].
Compared to other neutral atoms, the blackbody radiation shifts at room
temperature of $2s2p~{}^{1}S_{0}$ $\rightarrow$ $2s^{2}~{}^{3}P_{0}$ clock
transition is about 1.7$\times$10-17 [28], which is one or two orders of
magnitude smaller than that of the Mg, Sr and Yb atoms [29, 30, 31]. The
transition wavelength, 454.997 nm, lies in the optical range and the natural
line width is about 0.068 Hz [28]. Consequently, the quality factor, the ratio
of the transition frequency to the natural line width, of this line is about
1$\times$1016 [28]. Magic wavelengths for simultaneous trapping of the ground
and metastable states were calculated by Mitroy [27]. As far as we know, there
are no reported values for the dynamic $M1$ and $E2$ polarizabilities as well
as hyperpolarizabilities for Be atoms. Therefore, there is also no theoretical
analysis of the influence of $E2$ and $M1$ interactions and nonlinear Stark
shifts on the accuracy of optical lattice clocks.

In this paper, the energy levels and $E1$ matrix elements of the ground state
and low-lying excited states for Be atoms are calculated using the
relativistic configuration interaction plus core polarization (RCICP) method.
The static and dynamic $E1$, $M1$ and $E2$ polarizabilities as well as the
hyperpolarizabilities of the $2s^{2}~{}^{1}S_{0}$ and $2s2p~{}^{3}P_{0}$
states are further determined. Then, the laser intensities and detunings of
the optical lattice to improve the fractional Stark shifts of the clock
transition to 10-19-10-20 are analyzed in detail. Atomic units ($m_{e}$ = 1,
$e$ = 1, $\hbar$ = 1) are used throughout the paper unless stated otherwise.
The speed of light is taken to be 137.035 999 1 in our calculations.

##  2 Theoretical method

The key strategy of the RCICP method is to partition a Be atom into a Be2+
core plus two valence electrons. The calculation is separated into three
steps. The first step involves a Dirac-Fock (DF) calculation on the core of
Be2+ ions. The second step is to obtain the single-electron wave function of
valence orbitals. The single-particle orbitals are written as a linear
combination of the S-spinors [32, 33, 34, 35] which can be regarded as a
relativistic generalization of the Slater-type orbitals. The third step is to
diagonalize the Hamiltonian matrix in two-valence-electrons configuration
space. The effective Hamiltonian of the two valence electrons is written as

$H=\sum\limits_{i=1}^{2}(c\bm{\alpha}\cdot\bm{p}+(\beta-1)c^{2}+V_{\rm{core}}(\bm{r_{i}}))+\frac{1}{r_{12}}+V_{\rm{p2}}(\bm{r_{1}},\bm{r_{2}}),$
(1)

where the summation part represents single-electron Hamiltonian, $\bm{\alpha}$
and $\beta$ are the $4\times 4$ Dirac matrices, $\bm{p}$ is the momentum
operator, $c$ is the speed of light, and $\bm{r}$ is the position vector of
the valence electron. Moreover, $V_{\rm{core}}(\bm{r})$ is given by

$V_{\rm{core}}(\bm{r})=-\frac{Z}{r}+V_{\rm{dir}}(\bm{r})+V_{\rm{exc}}(\bm{r})+V_{\rm{p1}}(\bm{r}).$
(2)

Here, $Z$ is atomic number, $r$ is the distance of the valence electron with
respect to the origin of the coordinates. $V_{\rm{dir}}(\bm{r})$ and
$V_{\rm{exc}}(\bm{r})$ denote the direct and exchange interactions of the
valence electron with the core electrons, respectively. The $\ell$\- and
$j$-dependent one-electron polarization potential $V_{\rm{p1}}(\bm{r})$ can be
written as [36]

$V_{\rm{p1}}(\bm{r})=-\sum_{k=1}^{3}\frac{\alpha_{\rm{core}}^{k}}{2r^{2(k+1)}}\sum_{\ell,j}g_{\ell,j}^{2}(r)|\ell
j\rangle\langle\ell j|.$ (3)

The two-electron polarization potential is written as [36]

$V_{p2}(\bm{r}_{1},\bm{r}_{2})=-\sum_{k=1}^{3}\frac{\alpha_{\mathrm{core}}^{(k)}}{r_{1}^{2(k+1)}r_{2}^{2(k+1)}}\sum_{\ell,j}g_{k,\ell,j}(r_{1})g_{k,\ell,j}(r_{2})(\bm{r}_{1}\cdot\bm{r}_{2}),$
(4)

where $\ell$ and $j$ are the orbital and total angular momenta, respectively.
$\alpha_{\rm{core}}^{k}$ is the $k$th-order static polarizabilities of the
core electrons ($\alpha_{\mathrm{core}}^{(1)}$ = 5.227$\times 10^{-2}$ a.u.,
$\alpha_{\mathrm{core}}^{(2)}$ = 1.532$\times 10^{-2}$ a.u. and
$\alpha_{\mathrm{core}}^{(3)}$ = 1.135$\times 10^{-2}$ a.u. [37] for Be2+
ions), and $g_{k,\ell,j}^{2}(r)=1-\exp(-r^{2(k+2)}/\rho_{\ell,j}^{2(k+2)})$.
The cutoff parameters $\rho_{\ell,j}$ are tuned to reproduce the binding
energies of the ground state and some low-lying excited states, which are
listed in Table 1. The effective Hamiltonian of the valence electron is
diagonalized within a large S-spinor and L-spinor basis [35, 38]. L-spinors
can be regarded as a relativistic generalization of the Laguerre-type
orbitals.

The energy levels of some low-lying states of Be atoms are listed in Table 2,
and compared with the National Institute of Science and Technology (NIST)
tabulation [28]. It can be found that the present RCICP results are in good
agreement with those from the NIST tabulation, and the difference is no more
than 0.07%.

Table 1:  The radial cutoff parameter $\rho_{\ell,j}$ of the polarization potential of Be atoms. States |  $j$ |  $\rho_{\ell,j}$ (a. u.)
---|---|---
$s$ |  1/2 |  0.9587
$p$ |  1/2 |  0.8695
|  3/2 |  0.8672
$d$ |  3/2 |  1.3305
|  5/2 |  1.3286
Table 2:  Energy levels (cm-1) for some of the low-lying states of the Be atoms, which are given relatively to the core Be2+. The relative differences between the present RCICP results and the National Institute of Science and Technology (NIST) tabulation [28] are listed as Diff. State |  RCICP |  NIST [28] |  Diff.
---|---|---|---
$2s^{2}$ ${}^{1}S_{0}$ |  $-$222065.82 |  $-$222075.47 |  $-$0.004%
$2s3s$ ${}^{1}S_{0}$ |  $-$167384.93 |  $-$167398.13 |  $-$0.008%
$2p^{2}$ ${}^{3}P_{0}$ |  $-$162326.52 |  $-$162381.88 |  $-$0.034%
$2s4s$ ${}^{1}S_{0}$ |  $-$156783.08 |  $-$156830.05 |  $-$0.030%
$2s5s$ ${}^{1}S_{0}$ |  $-$152676.20 |  $-$152753.18 |  $-$0.050%
$2s6s$ ${}^{1}S_{0}$ |  $-$150653.17 |  $-$150754.23 |  $-$0.067%
$2s2p$ ${}^{3}P_{0}$ |  $-$200086.53 |  $-$200097.17 |  $-$0.005%
$2s3p$ ${}^{3}P_{0}$ |  $-$163129.57 |  $-$163168.01 |  $-$0.024%
$2s4p$ ${}^{3}P_{0}$ |  $-$155212.55 |  $-$155263.60 |  $-$0.033%
$2s5p$ ${}^{3}P_{0}$ |  $-$151967.42 |  $-$152010.08 |  $-$0.028%
$2s6p$ ${}^{3}P_{0}$ |  $-$150304.45 |  $-$150344.84 |  $-$0.027%
$2s2p$ ${}^{3}P_{1}$ |  $-$200089.21 |  $-$200096.55 |  $-$0.004%
$2s2p$ ${}^{1}P_{1}$ |  $-$179482.17 |  $-$179510.03 |  $-$0.016%
$2s3p$ ${}^{3}P_{1}$ |  $-$163109.39 |  $-$163168.01 |  $-$0.036%
$2s3p$ ${}^{1}P_{1}$ |  $-$161806.87 |  $-$161888.04 |  $-$0.050%
$2s4p$ ${}^{3}P_{1}$ |  $-$155183.49 |  $-$155263.61 |  $-$0.052%
$2s4p$ ${}^{1}P_{1}$ |  $-$154952.51 |  $-$155040.69 |  $-$0.057%
$2s5p$ ${}^{3}P_{1}$ |  $-$151947.39 |  $-$152010.08 |  $-$0.041%
$2s5p$ ${}^{1}P_{1}$ |  $-$151878.95 |  $-$151954.90 |  $-$0.050%
$2s2p$ ${}^{3}P_{2}$ |  $-$200089.78 |  $-$200094.21 |  $-$0.002%
$2s3p$ ${}^{3}P_{2}$ |  $-$163118.88 |  $-$163167.64 |  $-$0.030%
$2s4p$ ${}^{3}P_{2}$ |  $-$155199.36 |  $-$155263.60 |  $-$0.041%
$2s4f$ ${}^{3}F_{2}$ |  $-$153781.72 |  $-$153834.44 |  $-$0.034%
$2s3s$ ${}^{3}S_{1}$ |  $-$169987.75 |  $-$169994.54 |  $-$0.004%
$2p^{2}$ ${}^{3}P_{1}$ |  $-$162364.74 |  $-$162380.49 |  $-$0.010%
$2s3d$ ${}^{3}D_{1}$ |  $-$159976.31 |  $-$160021.74 |  $-$0.028%
$2s4s$ ${}^{3}S_{1}$ |  $-$157510.31 |  $-$157569.00 |  $-$0.037%
$2s4d$ ${}^{3}D_{1}$ |  $-$154082.45 |  $-$154133.85 |  $-$0.033%
$2s5s$ ${}^{3}S_{1}$ |  $-$152992.49 |  $-$153065.31 |  $-$0.048%
$2s5d$ ${}^{3}D_{1}$ |  $-$151418.88 |  $-$151471.72 |  $-$0.035%
$2p^{2}$ ${}^{1}D_{2}$ |  $-$165187.86 |  $-$165192.93 |  $-$0.003%
$2p^{2}$ ${}^{3}P_{2}$ |  $-$162369.64 |  $-$162378.50 |  $-$0.005%
$2s3d$ ${}^{3}D_{2}$ |  $-$159994.17 |  $-$160021.74 |  $-$0.017%
$2s3d$ ${}^{1}D_{2}$ |  $-$157576.19 |  $-$157647.08 |  $-$0.045%
$2s4d$ ${}^{3}D_{2}$ |  $-$154056.36 |  $-$154133.83 |  $-$0.050%
$2s4d$ ${}^{1}D_{2}$ |  $-$153194.25 |  $-$153294.53 |  $-$0.065%

##  3 Results and discussion

###  3.1 $E1$, $E2$ and $M1$ transitions matrix elements

The $E1$ transition matrix elements are calculated with a modified dipole
transition operator given by [39, 40, 41, 42, 43]

$\bm{D}=\bm{r}-[1-\mathrm{exp}(-\frac{r^{6}}{\rho^{6}})]^{1/2}\frac{\alpha_{\rm{core}}^{1}\bm{r}}{r^{3}}.$
(5)

The cutoff parameter $\rho$ is 1.0522 a.u., generated as
$\rho=\frac{1}{6}(2\rho_{s_{1/2}}+\rho_{p_{1/2}}+\rho_{p_{3/2}}+\rho_{d_{3/2}}+\rho_{d_{5/2}})$.

Table 3 lists the presently calculated $E1$ reduced matrix elements for
transitions between some low-lying states, along with a comparison with some
available theoretical [44, 45, 46, 47, 48, 49] and experimental results [50,
51, 52, 53]. For the $2s^{2}~{}^{1}S_{0}\rightarrow 2s2p~{}^{1}P_{1}$ resonant
transition, which is dominant contributing to the polarizability of the ground
state, the present RCICP result is in good agreement with experimental [50,
51] and theoretical [44, 45, 46, 47, 48, 49] results. The differences are less
than 1.5%. For $2s2p~{}^{3}P_{0}\rightarrow 2p^{2}~{}^{3}P_{1}$ and
$2s2p~{}^{3}P_{0}\rightarrow 2s3d~{}^{3}D_{1}$ transitions, the reduced matrix
elements are larger than 1.0 a.u., and the present RCICP results agree very
well with the experimental results [52, 53]. The agreement is better than 1%.
Moreover, there are no comparable experimental values for the transitions of
reduced matrix elements of less than 1.0 a.u., except for the
$2s2p~{}^{3}P_{0}\rightarrow 2s3s~{}^{3}S_{1}$ transition. The present results
agree well with other theoretical results [44, 45, 46, 47, 48, 49]. For
$2s2p~{}^{3}P_{0}\rightarrow 2s3s~{}^{3}S_{1}$ transition, the difference
between the present result and experimental results [52, 53] is no more than
4%.

Table 3:  Reduced $E1$ matrix elements (a.u.) for some of principal transitions of Be atoms. Transition | This work |  CICP [44] |  BCICP [45] |  BCIBP [46] |  TDGI [47] |  MCHF [48] |  HFR [49] | Exp.
---|---|---|---|---|---|---|---|---
$2s^{2}~{}^{1}S_{0}\rightarrow 2s2p~{}^{1}P_{1}$ | 3.2606 | 3.2597 | 3.260 | 3.262 | 3.270 | 3.256 | 3.306 |  3.22(6) [50]
|  |  |  |  |  |  |  |  3.29(5) [51]
$2s^{2}~{}^{1}S_{0}\rightarrow 2s3p~{}^{1}P_{1}$ | 0.2212 | 0.2179 | 0.222 | 0.222 |  | 0.221 |  |
$2s^{2}~{}^{1}S_{0}\rightarrow 2s4p~{}^{1}P_{1}$ | 0.0336 |  | 0.034 |  |  | 0.024 |  |
$2s^{2}~{}^{1}S_{0}\rightarrow 2s5p~{}^{1}P_{1}$ | 0.0620 |  | 0.062 |  |  | 0.057 |  |
$2s2p~{}^{3}P_{0}\rightarrow 2s3s~{}^{3}S_{1}$ | 0.9505 | 0.9091 | 0.948 | 0.961 | 0.534 | 0.961 | 0.954 |  0.97(1) [52]
|  |  |  |  |  |  |  |  0.99(2) [53]
$2s2p~{}^{3}P_{0}\rightarrow 2p^{2}~{}^{3}P_{1}$ | 1.9804 | 1.9740 |  | 1.972 |  | 1.972 | 2.045 |  1.96(1) [53]
$2s2p~{}^{3}P_{0}\rightarrow 2s3d~{}^{3}D_{1}$ | 1.5576 | 1.5551 | 1.556 | 1.568 | 1.118 | 1.568 | 1.503 |  1.54(2) [52]
|  |  |  |  |  |  |  |  1.54(2) [53]
$2s2p~{}^{3}P_{0}\rightarrow 2s4s~{}^{3}S_{1}$ | 0.2995 |  | 0.300 |  |  | 0.300 | 0.288 |
$2s2p~{}^{3}P_{0}\rightarrow 2s4d~{}^{3}D_{1}$ | 0.8276 |  | 0.827 |  |  | 0.829 | 0.737 |
Table 4:  The $M1$ and $E2$ matrix elements (a.u.) for principal transitions of Be atoms. The notation a[b] means a$\times 10^{b}$. Type | Transition | This work | Other studies | Type | Transition | This work | Other studies
---|---|---|---|---|---|---|---
M1 | $2s^{2}~{}^{1}S_{0}\rightarrow 2s3s~{}^{3}S_{1}$ |  1.24[$-$5] |  1.29[$-$5] [54] | M1 | $2s2p~{}^{3}P_{0}\rightarrow 2s2p~{}^{3}P_{1}$ | 1.43 |  1.41 [28]
M1 | $2s^{2}~{}^{1}S_{0}\rightarrow 2s4s~{}^{3}S_{1}$ |  6.77[$-$6] |  6.70[$-$6] [54] | M1 | $2s2p~{}^{3}P_{0}\rightarrow 2s2p~{}^{1}P_{1}$ |  1.16[$-$4] |  1.14[$-$4] [28]
M1 | $2s^{2}~{}^{1}S_{0}\rightarrow 2s5s~{}^{3}S_{1}$ |  4.51[$-$6] |  4.45[$-$6] [54] | M1 | $2s2p~{}^{3}P_{0}\rightarrow 2s3p~{}^{3}P_{1}$ |  5.24[$-$5] |
M1 | $2s^{2}~{}^{1}S_{0}\rightarrow 2s6s~{}^{3}S_{1}$ |  3.52[$-$6] |  3.33[$-$6] [54] | M1 | $2s2p~{}^{3}P_{0}\rightarrow 2s3p~{}^{1}P_{1}$ |  9.71[$-$5] |
E2 | $2s^{2}~{}^{1}S_{0}\rightarrow 2p^{2}~{}^{1}D_{2}$ | 6.17 |  | E2 | $2s2p~{}^{3}P_{0}\rightarrow 2s2p~{}^{3}P_{2}$ | 7.25 |  7.20 [28]
E2 | $2s^{2}~{}^{1}S_{0}\rightarrow 2p^{2}~{}^{3}P_{2}$ |  2.02[$-$1] |  | E2 | $2s2p~{}^{3}P_{0}\rightarrow 2s3p~{}^{3}P_{2}$ | 5.43 |
E2 | $2s^{2}~{}^{1}S_{0}\rightarrow 2s3d~{}^{3}D_{2}$ | 3.15 |  | E2 | $2s2p~{}^{3}P_{0}\rightarrow 2s4p~{}^{3}P_{2}$ | 2.04 |
E2 | $2s^{2}~{}^{1}S_{0}\rightarrow 2s3d~{}^{1}D_{2}$ | 12.48 |  12.50 [55] | E2 | $2s2p~{}^{3}P_{0}\rightarrow 2s4f~{}^{3}F_{2}$ | 3.05 |

Table 4 lists the presently calculated $E2$ and $M1$ matrix elements for some
important transitions and some available theoretical results [55, 54, 28]. The
$M1$ matrix elements are very small, about 10-4 $\sim$ 10-6, except for the
$2s2p~{}^{3}P_{0}\rightarrow 2s2p~{}^{3}P_{1}$ transition. The present RCICP
results agree very well with the other theoretical results [55, 28]. The
differences are less than 2%. For the $2s2p~{}^{3}P_{0}\rightarrow
2s2p~{}^{3}P_{1}$ transition, the $M1$ matrix element are four to six orders
of magnitude larger than the $M1$ matrix elements of the other $M1$
transitions. For the $E2$ matrix elements, we only found two available
theoretical data [55, 28], that is for the $2s^{2}~{}^{1}S_{0}\rightarrow
2s3d~{}^{1}D_{2}$ and $2s2p~{}^{3}P_{0}\rightarrow 2s2p~{}^{3}P_{2}$
transitions, to compare with the present results. The present RCICP results
agree very well with these two results, and the difference is no more than 1%.

###  3.2 Static and dynamic $E1$ polarizabilities

The dynamic $E1$ polarizability of the state $i$ with the total angular
momenta $j=0$ can be given by

$\alpha_{i}(\omega)=\sum_{n}\frac{f_{i\rightarrow n}}{\Delta E_{n\rightarrow
i}^{2}-\omega^{2}},$ (6)

where $\Delta E_{n\rightarrow i}$ is the transition energy and $\omega$ is the
laser frequency. When $\omega$ = 0, Eq. (6) is reduced to the static
polarizabilities. The $E1$ oscillator strength $f_{i\rightarrow n}$ is defined
as

$f^{(1)}_{i\to
n}=\frac{2|\langle\beta_{n}j_{n}\|\bm{D}\|\beta_{i}j_{i}\rangle|^{2}\Delta
E_{n\to i}}{3(2j_{i}+1)},$ (7)

where $j$ is the total angular momenta and $\beta$ represents all additional
angular momenta in addition to the total angular momenta $j$.

Table 5:  Static $E1$ polarizabilities $\alpha$ (a.u.) of the $2s^{2}~{}^{1}S_{0}$ and $2s2p~{}^{3}P_{0}$ states and the breakdown of the contributions of individual transitions for Be atoms. $2s^{2}~{}^{1}S_{0}$ |  |  $2s2p~{}^{3}P_{0}$
---|---|---
Contributions |  $\alpha$ |  |  Contributions |  $\alpha$
$2s2p~{}^{1}P_{1}$ |  36.5299 |  |  $2s3s~{}^{3}S_{1}$ |  4.3919
$2s3p~{}^{1}P_{1}$ |  0.1189 |  |  $2p^{2}~{}^{3}P_{1}$ |  15.2131
$2s4p~{}^{1}P_{1}$ |  0.0025 |  |  $2s3d~{}^{3}D_{1}$ |  8.8504
$2s5p~{}^{1}P_{1}$ |  0.0080 |  |  $2s4s~{}^{3}S_{1}$ |  0.3082
|  |  |  $2s4d~{}^{3}D_{1}$ |  2.1782
Remainds |  1.0757 |  |  |  8.1180
Core |  0.0523 |  |  |  0.0523
Total |  37.7873 |  |  |  39.1121
MCDHF [56] |  37.614 |  |  |  39.249
MCHF [57] |  37.62 |  |  |  39.33
CICP [27] |  37.73 |  |  |  39.04
TDGI [58, 59] |  37.62 |  |  |  36.08
Sum-over-states [60] |  36.6 |  |  |
Model potential [61] |  37.9 |  |  |
CI+MBPT [62] |  37.76 |  |  |
Hylleraas Weinhold [63] |  37.755 |  |  |
ECG [64] |  37.755 |  |  |
CI [65] |  37.8066 |  |  |
RCC [66] |  37.80(47) |  |  |
RCC+MBPT [67] |  37.86(17) |  |  |
Semi-empirical [68] |  37.69 |  |  |
Ab initio [69] |  38.12 |  |  |
VP+CI [70] |  37.59 |  |  |

Table 5 lists the presently calculated static $E1$ polarizabilities of the
$2s^{2}~{}^{1}S_{0}$ and $2s2p~{}^{3}P_{0}$ states and the breakdowns of the
contributions of individual transitions, along with a comparison with some
available theoretical results [56, 57, 27, 58, 59, 60, 61, 62, 63, 64, 65, 66,
67, 68, 69]. We can find that the polarizability of the $2s^{2}~{}^{1}S_{0}$
state is dominated by the $2s^{2}~{}^{1}S_{0}\rightarrow 2s2p~{}^{1}P_{1}$
transition, while for the $2s2p~{}^{3}P_{0}$ state is dominated by the
$2s2p~{}^{3}P_{0}\rightarrow 2s3s~{}^{3}S_{1}$, $2s2p~{}^{3}P_{0}\rightarrow
2p^{2}~{}^{3}P_{1}$, and $2s2p~{}^{3}P_{0}\rightarrow 2s3d~{}^{3}D_{1}$
transitions. The “Remains” in the table represents the contributions from
highly excited bound and continuum states of the valence electrons. The “Core”
denotes the contributions of the core ($1s^{2}$) electrons, which is
calculated by using a pseudospectral oscillator strength distribution [71, 68,
72]. The present total polarizability is in good agreement with other
theoretical results [56, 57, 27, 58, 59, 61, 62, 63, 64, 65, 66, 67, 68, 69,
70], and the difference is no more than 1%.

Fig. 1 depicts the dynamic polarizabilities of the $2s^{2}~{}^{1}S_{0}$ and
$2s2p~{}^{3}P_{0}$ states. Two magic wavelengths are found which are
identified with arrows. One of them, 300.03 nm, lies between the resonant
transitions of $2s2p~{}^{3}P_{0}\rightarrow 2s3s~{}^{3}S_{1}$ and
$2s2p~{}^{3}P_{0}\rightarrow 2p^{2}~{}^{3}P_{1}$. Another one, 252.28 nm, is
located near the resonant wavelength of $2s2p~{}^{3}P_{0}\rightarrow
2p^{2}~{}^{3}P_{1}$ transition. The present results are in good agreement with
the calculations from Ref. [27], 300.2 and 252.3 nm. Here, we recommend that
the 300.03-nm magic wavelength can be used for magic-wavelength trapping in
the experiment, since this magic wavelength has a 30-nm difference from the
resonant wavelength of $2s2p~{}^{3}P_{0}\rightarrow 2s3s~{}^{3}S_{1}$
transition. However, the 252.28-nm magic wavelength is only a 10-nm difference
from the resonant wavelength of $2s2p~{}^{3}P_{0}\rightarrow
2p^{2}~{}^{3}P_{1}$ transition, and it is far away from the visible region.
Therefore, it is the best choice to use the 300.03-nm magic wavelength for
magic-wavelength trapping in experiments.

Figure 1: Dynamic $E1$ polarizabilities (a.u.) of the $2s^{2}~{}^{1}S_{0}$ and
$2s2p~{}^{3}P_{0}$ states of Be atoms. Two magic wavelengths are identified
with arrows.

Table 6 lists the contributions of individual transitions to the dynamic $E1$
polarizabilities of the $2s^{2}~{}^{1}S_{0}$ and $2s2p~{}^{3}P_{0}$ states at
the magic wavelength 300.03 nm. The polarizability of the $2s^{2}~{}^{1}S_{0}$
state is dominated by the resonant $2s^{2}~{}^{1}S_{0}\rightarrow
2s2p~{}^{1}P_{1}$ transition, the contribution is more than 98%, while the
polarizability of the $2s2p~{}^{3}P_{0}$ state is dominated by the
$2s2p~{}^{3}P_{0}\rightarrow 2s3s~{}^{3}S_{1}$, $2s2p~{}^{3}P_{0}\rightarrow
2p^{2}~{}^{3}P_{1}$, and $2s2p~{}^{3}P_{0}\rightarrow 2s3d~{}^{3}D_{1}$
transitions. The contribution of $2s2p~{}^{3}P_{0}\rightarrow
2s3s~{}^{3}S_{1}$ transition is negative.

Table 6:  Breakdowns of the contributions of individual transitions to the dynamic $E1$ polarizabilities (a.u.) of the $2s^{2}~{}^{1}S_{0}$ and $2s2p~{}^{3}P_{0}$ states at the 300.03 nm magic wavelength. $2s^{2}~{}^{1}S_{0}$ |  |  $2s2p~{}^{3}P_{0}$
---|---|---
Contributions |  $\alpha$ |  |  Contributions |  $\alpha$
$2s2p~{}^{1}P_{1}$ |  94.6108 |  |  $2s3s~{}^{3}S_{1}$ |  $-$19.4274
$2s3p~{}^{1}P_{1}$ |  0.1846 |  |  $2p^{2}~{}^{3}P_{1}$ |  69.0677
|  |  |  $2s3d~{}^{3}D_{1}$ |  28.7577
|  |  |  $2s4d~{}^{3}D_{1}$ |  4.6003
Remainds |  1.1508 |  |  |  12.9479
Core |  0.0523 |  |  |  0.0523
Total |  95.9985 |  |  |  95.9985

###  3.3 Static $M1$ and $E2$ polarizabilities as well as
hyperpolarizabilities

The static $M1$ and $E2$ polarizabilities for the state $i$ can be given by
[73]

$\alpha^{M1}=\frac{2}{3(2j_{i}+1)}\sum\limits_{n}\frac{|\langle\beta_{n}j_{n}\|\bm{M1}\
|\beta_{i}j_{i}\rangle|^{2}}{\Delta E_{n\rightarrow i}},$ (8)
$\alpha^{E2}=\frac{2}{5(2j_{i}+1)}\sum\limits_{n}\frac{|\langle\beta_{n}j_{n}\|\bm{Q}\
|\beta_{i}j_{i}\rangle|^{2}}{\Delta E_{n\rightarrow i}},$ (9)

where $\bm{M1}$ and $\bm{Q}$ are the magnetic-dipole and electric-quadrupole
transition operators, respectively.

The dynamic hyperpolarizabilities $\gamma^{l}(\omega)$ and
$\gamma^{c}(\omega)$ under the linearly and circularly polarized lights for
the $j_{i}=0$ state can be written as, respectively [24, 74]

$\gamma^{l}(\omega)=\frac{1}{9}\mathcal{T}(1,0,1,\omega,-\omega,\omega)+\frac{2}{45}\mathcal{T}(1,2,1,\omega,-\omega,\omega),$
(10)
$\gamma^{c}(\omega)=\frac{1}{9}\mathcal{T}(1,0,1,\omega,-\omega,\omega)+\frac{1}{90}\mathcal{T}(1,2,1,\omega,-\omega,\omega).$
(11)

Where $\mathcal{T}(1,0,1,\omega,-\omega,\omega)$ and
$\mathcal{T}(1,2,1,\omega,-\omega,\omega)$ are expressed as the following
general formula [74]

$\displaystyle\mathcal{T}(j_{a},j_{b},j_{c},\omega,-\omega,\omega)=4\sum\limits_{\beta_{a},\beta_{b},\beta_{c}}\langle\beta_{i}j_{i}||D||\beta_{a}j_{a}\rangle\langle\beta_{a}j_{a}||D||\beta_{b}j_{b}\rangle\langle\beta_{b}j_{b}||D||\beta_{c}j_{c}\rangle\langle\beta_{c}j_{c}||D||\beta_{i}j_{i}\rangle$
$\displaystyle\times\\{\frac{1}{(\Delta E_{a\rightarrow i}-\omega)(\Delta
E_{b\rightarrow i}-2\omega)(\Delta E_{c\rightarrow
i}-\omega)}+\frac{1}{(\Delta E_{a\rightarrow i}+\omega)(\Delta E_{b\rightarrow
i}+2\omega)(\Delta E_{c\rightarrow i}+\omega)}$ $\displaystyle+\frac{4\Delta
E_{a\rightarrow i}\Delta E_{c\rightarrow i}}{(\Delta E_{a\rightarrow
i}+\omega)(\Delta E_{a\rightarrow i}-\omega)\Delta E_{b\rightarrow i}(\Delta
E_{c\rightarrow i}+\omega)(\Delta E_{c\rightarrow
i}-\omega)}\\}+8(-1)^{j_{a}+j_{c}+1}\delta(j_{b},j_{i})$
$\displaystyle\times\sum\limits_{\beta_{a}}\frac{\Delta E_{a\rightarrow
i}\left|\langle\beta_{i}j_{i}||D||\beta_{a}j_{a}\rangle\right|^{2}}{\Delta
E_{a\rightarrow i}^{2}-\omega^{2}}\sum\limits_{\beta_{c}}\frac{(3\Delta
E_{c\rightarrow
i}^{2}+\omega^{2})\left|\langle\beta_{i}j_{i}||D||\beta_{c}j_{c}\rangle\right|^{2}}{(\Delta
E_{c\rightarrow i}^{2}-\omega^{2})^{2}},$ (12)

where $\sum\limits_{\beta_{a},\beta_{b},\beta_{c}}$ represents three
summations over a large number of intermediate states. When $\omega$ = 0, Eqs.
(10) and (11) are reduced to the static hyperpolarizabilities.

Table 7 lists the presently calculated the static $M1$ and $E2$
polarizabilities for the $2s^{2}~{}^{1}S_{0}$ and $2s2p~{}^{3}P_{0}$ states.
These polarizabilities are compared with some available theoretical results
[64, 75, 62, 58, 70, 76]. For the $E2$ polarizability of the
$2s^{2}~{}^{1}S_{0}$ state, the present result is in excellent agreement with
the calculation of the explicitly correlated Gaussian (ECG) basis [64], the
CICP [75], and the configuration interaction approach and many-body
perturbation theory (CI+MBPT) [62]. The difference is no more than 0.15%.
There are no other theoretical or experimental values available for the $E2$
polarizability of the $2s2p~{}^{3}P_{0}$ state and the $M1$ polarizabilities
of the $2s^{2}~{}^{1}S_{0}$ and $2s2p~{}^{3}P_{0}$ states.

Table 7:  Static $M1$ and $E2$ polarizabilities (a.u.) for the $2s^{2}~{}^{1}S_{0}$ and $2s2p~{}^{3}P_{0}$ states. Methods |  $2s^{2}~{}^{1}S_{0}$ |  |  $2s2p~{}^{3}P_{0}$
---|---|---|---
$\alpha^{E2}$ |  $\alpha^{M1}$ |  |  $\alpha^{E2}$ |  $\alpha^{M1}$
This work |  300.98 |  3.68$\times$ 10-7 |  |  1.58$\times$ 106 |  4.87$\times$ 105
ECG [64] |  300.96 |  |  |  |
CICP [75] |  300.7 |  |  |  |
CI+MBPT [62] |  300.6(3) |  |  |  |
TDG1 [58] |  285.6 |  |  |  |
VP+CI [70] |  299.4 |  |  |  |
CCD+ST [76] |  298.8 |  |  |  |

Table 8 presents the static hyperpolarizabilities of the $2s^{2}~{}^{1}S_{0}$
and $2s2p~{}^{3}P_{0}$ states. We find that the static hyperpolarizabilities
are dominated by the $\mathcal{T}(1,2,1)$ term. For the $2s^{2}~{}^{1}S_{0}$
state, the present RCICP result is in good agreement with other theoretical
results [77, 76, 78, 79, 80, 81, 82, 83]. There are no other theoretical
values for the $2s2p~{}^{3}P_{0}$ state available for comparison.

Table 8:  The static hyperpolarizabilities $\gamma$ (a.u.) of the $2s^{2}~{}^{1}S_{0}$ and $2s2p~{}^{3}P_{0}$ states. The notation a[b] means a$\times 10^{b}$. Methods |  Contributions |  $2s^{2}~{}^{1}S_{0}$ |  $2s2p~{}^{3}P_{0}$
---|---|---|---
|  $\frac{1}{9}\mathcal{T}(1,0,1)$ |  6.081[3] |  4.904[4]
|  $\frac{2}{45}\mathcal{T}(1,2,1)$ |  2.963[4] |  2.628[5]
This work |  |  3.571[4] |  3.118[5]
FCI [77] |  |  2.7227[4] |
CCD+ST [76] |  |  3.148[4 |
MP4 [80] |  |  3.1[4] |
CC-R12 [78] |  |  3.21[4] |
ECG [79] |  |  3.0989[4] |
RHF [81] |  |  3.95[4] |
MCSCF [82] |  |  3.930[4] |
FD HF [83] |  |  3.912[4] |

###  3.4 Dynamic $E2$ and $M1$ polarizabilities as well as dynamic
hyperpolarizabilities around magic wavelength.

The dynamic $M1$ and $E2$ polarizabilities for the $j_{i}=0$ state can be
given by [73]

$\alpha^{M1}(\omega)=\frac{2}{3}\sum\limits_{n}\frac{\Delta E_{n\rightarrow
i}|\langle\beta_{n}j_{n}\|\bm{M1}\ |\beta_{i}j_{i}\rangle|^{2}}{\Delta
E^{2}_{n\rightarrow i}-\omega^{2}},$ (13)

and

$\alpha^{E2}(\omega)=\frac{1}{30}(\alpha\omega)^{2}\sum\limits_{n}\frac{\Delta
E_{n\rightarrow
i}|\langle\beta_{n}j_{n}\|\bm{Q}\|\beta_{i}j_{i}\rangle|^{2}}{\Delta
E^{2}_{n\rightarrow i}-\omega^{2}},$ (14)

where $\alpha$ in Eq. (14) is the fine structure constant.

Table 9 lists the presently calculated dynamic $M1$ and $E2$ polarizabilities
for the $2s^{2}~{}^{1}S_{0}$ and $2s2p~{}^{3}P_{0}$ states at the 300.03-nm
magic wavelength. As can be seen from the table, the absolute value of $M1$
polarizability of the $2s^{2}~{}^{1}S_{0}$ state is five orders of magnitude
smaller than $2s2p~{}^{3}P_{0}$ state, and the $M1$ polarizability of the
$2s2p~{}^{3}P_{0}$ state is negative. Thus, the differential $M1$
polarizability ($\Delta\alpha^{M1}(\omega)$) between these two states is
determined by the $2s2p~{}^{3}P_{0}$ state. The differential $E2$
polarizability ($\Delta\alpha^{E2}(\omega)$) is one order of magnitude larger
than that of the $\Delta\alpha^{M1}(\omega)$. Therefore, the differential
dynamic multipolar polarizability
($\Delta\alpha^{QM}(\omega)=\Delta\alpha^{M1}(\omega)+\Delta\alpha^{E2}(\omega)$)
is mainly determined by the $\Delta\alpha^{E2}(\omega)$.

Table 9:  The dynamic $M1$ and $E2$ polarizabilities (a.u.) for the $2s^{2}~{}^{1}S_{0}$ and $2s2p~{}^{3}P_{0}$ states at the 300.03-nm magic wavelength. The dynamic multipolar polarizability $\alpha^{QM}(\omega)=\alpha^{M1}(\omega)+\alpha^{E2}(\omega)$. The $\Delta\alpha$ represents the differential polarizabilities between these two states. The notation a[b] means a$\times 10^{b}$. Polarizabilities |  $2s^{2}~{}^{1}S_{0}$ |  $2s2p~{}^{3}P_{0}$ |  $\Delta\alpha(2s^{2}~{}^{1}S_{0}\rightarrow 2s2p~{}^{3}P_{0})$
---|---|---|---
$\alpha^{M1}(\omega)$ |  2.38[$-$9] |  $-$1.47[$-$4] |  $-$1.47[$-$4]
$\alpha^{E2}(\omega)$ |  4.33[$-$4] |  2.85[$-$3] |  2.42[$-$3]
$\alpha^{QM}(\omega)$ |  4.33[$-$4] |  2.70[$-$3] |  2.27[$-$3]

Table 10 gives the dynamic hyperpolarizabilities of the $2s^{2}~{}^{1}S_{0}$
and $2s2p~{}^{3}P_{0}$ states and breakdowns of the contributions to the
dynamic hyperpolarizabilities at 300.03 nm magic wavelength. We found that the
dynamic hyperpolarizability $\gamma^{l}(\omega)$ of the $2s^{2}~{}^{1}S_{0}$
state in the linearly polarized light is four orders of magnitude smaller than
$2s2p~{}^{3}P_{0}$ state, and the $\gamma^{c}(\omega)$ of the
$2s^{2}~{}^{1}S_{0}$ state in the circularly polarized light is three orders
of magnitude smaller than the $2s^{2}~{}^{3}P_{0}$ state. Therefore, the
differential dynamic hyperpolarizabilities ($\Delta\gamma^{l}(\omega)$ and
$\Delta\gamma^{c}(\omega)$) in the linearly and circularly polarized lights
are determined by the $2s2p~{}^{3}P_{0}$ state.

Table 10:  The dynamic hyperpolarizabilities (a.u.) of the $2s^{2}~{}^{1}S_{0}$ and $2s2p~{}^{3}P_{0}$ states at the 300.03-nm magic wavelength. The $\Delta\gamma{(\omega)}$ represents the differential hyperpolarizabilities between these two states. The superscript $l$ and $c$ represent the linearly and circularly polarized lights. The notation a[b] means a$\times 10^{b}$. Contribution | $\gamma^{l}{(\omega)}$ | Contribution | $\gamma^{c}{(\omega)}$
---|---|---|---
$2s^{2}~{}^{1}S_{0}$ | $2s2p~{}^{3}P_{0}$ | $2s^{2}~{}^{1}S_{0}$ | $2s2p~{}^{3}P_{0}$
$\frac{1}{9}T(1,0,1,\omega,$-$\omega,\omega)$ |  $-$7.91[5] |  $-$3.01[6] |  $\frac{1}{9}T(1,0,1,\omega,$-$\omega,\omega)$ |  $-$7.91[5] |  $-$3.01[6]
$\frac{2}{45}T(1,2,1,\omega,$-$\omega,\omega)$ | 7.65[5] | 7.43[8] |  $\frac{1}{90}T(1,2,1,\omega,$-$\omega,\omega)$ | 1.91[5] | 1.86[8]
Total |  $-$2.59[4] | 7.40[8] |  |  $-$6.00[5] | 1.83[8]
$\Delta\gamma^{l}{(\omega)}$ | 7.40[8] |  | $\Delta\gamma^{c}{(\omega)}$ | 1.83[8] |
Figure 2:  Differential dynamic $M1$ and $E2$ polarizabilities (a.u.) as well
as hyperpolarizabilitiy (a.u.) around the 300.03-nm magic wavelength. (a-b)
The differential dynamic $M1$ and $E2$ polarizabilities between
$2s^{2}~{}^{1}S_{0}$ and $2s2p~{}^{3}P_{0}$ states. (c-d) The differential
dynamic hyperpolarizabilities between $2s^{2}~{}^{1}S_{0}$ and
$2s2p~{}^{3}P_{0}$ states in the linearly and circularly polarized lights.

Fig. 2 shows the differential dynamic $M1$ and $E2$ polarizabilities as well
as hyperpolarizabilities between $2s^{2}~{}^{1}S_{0}$ and $2s2p~{}^{3}P_{0}$
states around the 300.03-nm magic wavelength. These differential
polarizabilities are extremely important in analyzing multipole and nonlinear
Stark shifts.

###  3.5 Stark shifts near the operational magic conditions

For atoms trapped under a one-dimensional optical lattice with the laser
frequency $\omega$ and the linearly polarized laser field intensity $I$, the
Stark shift for a clock transition can be expressed as [20, 25]

$\displaystyle
h\Delta\nu=[\frac{\partial\Delta\alpha^{E1}(\omega)}{\partial\nu}\delta-\Delta\alpha^{QM}(\omega)](n_{z}+\frac{1}{2})\sqrt{\frac{E_{R}}{\alpha^{E1}(\omega)}}I^{1/2}$
$\displaystyle-[\frac{\partial\Delta\alpha^{E1}(\omega)}{\partial\nu}+\frac{3}{8}\frac{E_{R}\Delta\gamma^{l}(\omega)}{\alpha^{E1}(\omega)}(n_{z}^{2}+n_{z}+\frac{1}{2})]I$
$\displaystyle+\frac{1}{2}\Delta\gamma^{l}(\omega)\sqrt{\frac{E_{R}}{\alpha^{E1}(\omega)}}(n_{z}+\frac{1}{2})I^{3/2}-\frac{1}{4}\Delta\gamma^{l}(\omega)I^{2},$
(15)

where $\delta$ is the frequency detuning of the lattice laser frequency $\nu$
relative to the magic frequency $\nu_{m}=\omega_{m}/2\pi$, $n_{z}$ is the
vibrational state of atoms along the $z$ axis [18], and
$E_{R}=h^{2}/(2M\lambda_{magic}^{2})$ is the lattice photon recoil energy with
$M$ being the atomic mass.

Figure 3: The fractional Stark shifts $\left|\Delta\nu\right|/\nu_{0}$ of the
clock transition with the increase of laser intensity $I$ and detunings
$\delta$. The color gradients represent the different fractional Stark shifts.
The black solid lines indicate the contour line of the fractional Stark shifts
of 1.0 $\times$ 10-17. The blue solid lines indicate the contour line of
fractional Stark shifts of 1.0 $\times$ 10-18, and when the laser intensity
$I$ is in the range of 14.3 $\sim$ 15.9 kW/cm2 and detuning $\delta$ is in the
range of 40.7 $\sim$ 40.9 MHz, the fractional Stark shift is less than the
level of 1.0 $\times$ 10-18. The green solid lines in the illustration
represent the contour line of the fractional Stark shifts of 1.0 $\times$
10-19, and when the $I$ is in the range of 15.01 $\sim$ 15.46 kW/cm2 and
$\delta$ is in the range of 40.73 $\sim$ 40.76 MHz, the fractional Stark shift
is less than the level of 1.0 $\times$ 10-19.

Here, we assume that the Be atoms are trapped in the $n_{z}=0$ vibrational
state. The multipolar and nonlinear Stark shifts are obtained using Eq. (3.5).
Fig. 3 presents the fractional Stark shifts $\left|\Delta\nu\right|/\nu_{0}$
of clock transition, the ratio of absolute values of Stark shifts to clock
transition frequency, with the increase of laser intensity $I$ and detuning
$\delta$. The color gradients represent the different fractional Stark shifts.
The black solid lines indicate the contour line of fractional Stark shifts of
1.0 $\times$ 10-17, and the blue solid lines indicate the contour line of
fractional Stark shifts of 1.0 $\times$ 10-18. The green solid lines in the
illustration represent the contour line of fractional Stark shifts of 1.0
$\times$ 10-19. In order to reduce the multipolar and nonlinear Stark shifts
and make the Stark shifts insensitive to the $I$ and $\delta$, it should
choose the vertex of the contour lines to determine the position of the laser
intensity $I$ and detuning $\delta$. We find that when the $I$ is in the range
of 14.3 $\sim$ 15.9 kW/cm2 and $\delta$ is in the range of 40.7 $\sim$ 40.9
MHz, the fractional Stark shifts of the clock transition are lower than 1.0
$\times$ 10-18. While, when the $I$ is in the range of 15.01 $\sim$ 15.46
kW/cm2 and $\delta$ is in the range of 40.73 $\sim$ 40.76 MHz, the fractional
Stark shifts are lower than 1.0 $\times$ 10-19, as shown in the illustration
in Fig. 3. These distinctive conditions can provide a reference for the
development of the Be optical lattice clock at the level of 10-19.

##  4 Conclusions

The energy levels and $E1$ matrix elements of the low-lying states of Be atoms
have been calculated using the RCICP method. The static and dynamic $E1$,
$M1$, and $E2$ polarizabilities as well as hyperpolarizabilities of the
$2s^{2}~{}^{1}S_{0}$ and $2s2p~{}^{3}P_{0}$ states are determined. Then, two
magic wavelengths, 300.03 nm and 252.28 nm, of the
$2s^{2}~{}^{1}S_{0}\rightarrow 2s2p~{}^{3}P_{0}$ clock transition are found.
We recommend that the 300.03-nm magic wavelength can be used for magic
trapping. The $\Delta\alpha^{M1}(\omega)$ and $\Delta\alpha^{E2}(\omega)$ as
well as differential dynamic hyperpolarizabilities around the 300.03-nm magic
wavelength are determined. In addition, we find that the
$\Delta\alpha^{M1}(\omega)$ is determined by $2s2p~{}^{3}P_{0}$ state. The
$\Delta\alpha^{QM}(\omega)$ is mainly determined by the
$\Delta\alpha^{E2}(\omega)$. The differential dynamic hyperpolarizability in
the linearly and circularly polarized lights are all determined by the
$2s2p~{}^{3}P_{0}$ state.

Finally, the multipolar and nonlinear Stark shifts of the clock transition
near the magic wavelength are calculated in detail. We find that when the
laser intensity is in the range of 14.3 $\sim$ 15.9 kW/cm2 and $\delta$ is in
the range of 40.7 $\sim$ 40.9 MHz, the fractional Stark shifts of the clock
transition are lower than 1.0 $\times$ 10-18. While, when the $I$ is in the
range of 15.01 $\sim$ 15.46 kW/cm2 and $\delta$ is in the range of 40.73
$\sim$ 40.76 MHz, the fractional Stark shifts are lower than 1.0 $\times$
10-19. These will provide important support for developing ultra-high-
precision Be optical clocks.

## Acknowledgments

This work has been supported by the National Natural Science Foundation of
China under Grants No. 12174316, the Young Teachers Scientific Research
Ability Promotion Plan of Northwest Normal University (NWNU-LKQN2020-10) and
Funds for Innovative Fundamental Research Group Project of Gansu Province
(20JR5RA541).

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# Spatio-temporal observation of higher-order modulation instability in a
recirculating fiber loop

François Copie  Univ. Lille, CNRS, UMR 8523 - PhLAM - Physique des Lasers
Atomes et Molécules, F-59000 Lille, France  Corresponding author:
<EMAIL_ADDRESS>Pierre Suret  Univ. Lille, CNRS, UMR 8523 -
PhLAM - Physique des Lasers Atomes et Molécules, F-59000 Lille, France
Stephane Randoux  Univ. Lille, CNRS, UMR 8523 - PhLAM - Physique des Lasers
Atomes et Molécules, F-59000 Lille, France

###### Abstract

We experimentaly investigate higher-order seeded modulation instability in an
optical fiber experiment. The recirculating loop configuration with round-trip
losses compensation enables the observation in single-shot of the spatio-
temporal evolution of an initially modulated continuous field revealing
intricate yet deterministic dynamics. By tuning the modulation period, a
continuous transition between perfectly coherent and purely noise-driven
dynamics is observed that we characterise by means of a statistical study.

††journal: ol

Modulation Instability (MI) is one of the most ubiquitous phenomena of
nonlinear dynamics owing to its appearance in very diverse fields including
hydrodynamics, Bose-Einstein condensate physics or optics to name a few [1].
It occurs when a pump field interacts with a weaker seed field and manifests
itself, at early stage, by the exponential growth of the seed provided that
its frequency falls below a certain cutoff which defines the so-called MI gain
band. In optical fiber systems, one of the most common scenario encountered is
the noise-induced or spontaneous MI in which a single continuous wave (cw)
pump field perturbed by spectral noise breaks up into a partially stochastic
train of short pulses before experiencing a complex spatio-temporal dynamics
[2, 3]. Another common configuration is the coherently seeded MI which is
triggered by a weak modulation of a cw pump laser and that leads to the
formation of well defined pulse trains [4]. At longer propagation distance,
the dynamics triggered by seeded MI can feature quasi-periodic spatio-temporal
evolutions of the wavefield that have been linked to the Fermi-Pasta-Ulam-
Tsingou (FPUT) recurrences [5]. More specifically, such a parallel is drawn
only for ‘1st-order MI’ which is a subset of the seeded MI in which the
modulation frequency of the cw field lies in the outer half of the MI gain
band (See Fig. 1(c)). Conversely, so-called higher-order MI can arise from a
single modulation whose frequency falls below half of the cutoff frequency and
results in considerably different spatio-temporal dynamics of the wavefield
[6, 7, 8, 9]. Indeed, in that case one or several harmonics of the modulation
frequency also fall in the MI gain band generated by the pump which leads to
intricate multiple wave mixing. The early stage of the dynamics features a
generic splitting cascade that has been succesfully described analytically as
resulting from the nonlinear superposition of elementary breathers [7]. Yet,
the long term dynamics of higher-order MI remains largely unexplored.

Until relatively recently, MI has been the subject of numerous experimental
observations in fiber based systems mainly limited to measurements of averaged
quantities such as conventional optical spectra or autocorrelation traces for
instance due to the intrinsic rapid timescale associated to the process [2,
10]. In the last 10 years though, the development of single-shot detection
systems and distributed measurement methods based in particular on dispersive
Fourier transformation (DFT), time-lens systems and time-domain reflectometry
has attracted particular interest and enabled, among other things, the precise
characterisation of MI-driven ultrafast dynamics [11, 12, 13, 14, 15, 16, 17].
In the past few years, the recirculating optical fiber loop configuration
(historically developed to study long-haul transmissions of solitonic pulses
[18, 19]) has also proven to be a powerful experimental platform to observe
and investigate complex spatio-temporal dynamics induced by MI [20, 21, 22].

In this work, we report original observations of higher-order MI realised in a
recirculating optical fiber loop. We recorded the spatio-temporal evolution of
long square pulses sinusoidally modulated that propagate over hundreds of
kilometers, exhibiting a variety of remarkable scenarios. It generally
consists in a pulse splitting sequence followed by a complex coherent pulse
recombination dynamics. When changing the frequency of the initial modulation,
we were able to reveal a controlled interplay between higher-order seeded MI
and the noise-induced MI thanks to our single-shot recording technique.

The evolution of optical wavefields propagating in our recirculating fiber
loop is well described by the following 1D nonlinear Schrödinger (NLS)
equation including effective linear dissipation:

$i\frac{\partial\Psi}{\partial
z}=\frac{\beta_{2}}{2}\frac{\partial^{2}\Psi}{\partial
t^{2}}-\gamma|\Psi|^{2}\Psi-i\frac{\alpha_{\text{eff}}}{2}\Psi$ (1)

$\Psi(z,t)$ is the envelop of the field, $\beta_{2}$ and $\gamma$ are the
group velocity dispersion (GVD) and nonlinear coefficients at
$1550\text{\,}\mathrm{n}\mathrm{m}$ respectively, $\alpha_{\text{eff}}$ is the
effective power loss coefficient, $z$ is the dimensional propagation distance
and $t$ the time in the frame travelling at the group velocity of the pump
field. The fiber used in our work exhibits anomalous GVD at the pump
wavelength ($\beta_{2}<0$) which is refered to as the focusing regime [23].
For the parameters of our experiments, the MI gain curve given by $g(\Delta
f)=|\beta_{2}\Delta f|\sqrt{f_{c}^{2}-\Delta f^{2}}$, where
$f_{c}=\sqrt{(4\gamma P_{p})/|\beta_{2}|}/(2\pi)$ is the MI cutoff frequency,
is plotted in Fig. 1(c) [3]. $\Delta f$ is the pump-seed frequency detuning
which is varied in our experiments by changing the frequency of a seed laser
around that of a pump laser of power $P_{p}$. The power decay during
propagation (longitudinal decrease of $|\Psi|^{2}$) influences the MI gain
curve which varies along propagation. Note that the expression of $g(\Delta
f)$ is derived analytically in the case of an initially infinitely small
modulation, i.e. for very small seed power. In the presented results, the
power of the seed field is $9\text{\,}\mathrm{d}\mathrm{B}$ lower than that of
the pump field.

Figure 1: (a) Schematic of the experimental setup. (b) Evolution of the power
decay measured in an experimental run. (c) Small signal MI gain computed from
experimental parameters. $P_{p}\sim$24\text{\,}\mathrm{m}\mathrm{W}$$,
$\gamma=$1.23\text{\,}\mathrm{/}\mathrm{W}\mathrm{/}\mathrm{k}\mathrm{m}$$,
$\beta_{2}($1550\text{\,}\mathrm{n}\mathrm{m}$)=$-22\text{\,}\mathrm{p}\mathrm{s}^{2}\mathrm{/}\mathrm{k}\mathrm{m}$$,
$\alpha_{\text{eff}}=$3.36\text{\times}{10}^{-6}\text{\,}\mathrm{m}^{-1}$$.
EDFA: Erbium doped fiber amplifier, AOM: Acousto-optic modulator, PD:
Photodetector, WDM: Wavelength division multiplexer.

Our experimental setup is schematically depicted in Fig. 1(a). Two cw lasers
respectively called pump and seed are mixed via a 50/50 coupler before being
amplified by an Erbium doped fiber amplifier (EDFA). This gives a modulated
signal at a beating frequency given by the detuning between the two lasers.
This field is then modulated by an acousto-optic modulator (AOM) to generate
long pulses ($250\text{\,}\mathrm{n}\mathrm{s}$) which mitigate Brillouin
scattering. The pulses are then launched into an
$8\text{\,}\mathrm{k}\mathrm{m}$-long recirculating loop made of standard
single mode fiber. At each round-trip, 10% of the circulating power is
extracted and directed towards a photodetector (PD) coupled to a sampling
oscilloscope leading to an overall $32\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$
detection bandwidth. The losses accumulated over one circulation in the fiber
loop are partially compensated thanks to a counter-propagating Raman pump
coupled in and out of the loop via wavelength division multiplexers (WDMs).
This reduces the effective power decay rate of the circulating field to
$0.015\text{\,}\mathrm{d}\mathrm{B}\mathrm{/}\mathrm{k}\mathrm{m}$ or
equivalently
$\alpha_{\text{eff}}\sim$3.36\text{\times}{10}^{-6}\text{\,}\mathrm{m}^{-1}$$
(see Fig. 1(b)). Data recorded with the oscilloscope consist in a succession
of sequences (one per round-trip) that are numerically-processed to construct
single-shot real-time spatio-temporal diagrams of the wavefield dynamics. By
tuning the frequency of the seed around that of the pump, the system is set in
an initial state where different orders of MI can be observed.

We have recorded several hundreds of spatio-temporal diagrams covering the
full MI bandwidth by tuning the seed frequency with a step of
$\sim$125\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}$$. Figure 2 shows excerpts of
our experimental results obtained for five values of frequency detuning
ranging from -7.48 to $-0.7\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$. The
corresponding real-time spatio-temporal dynamics of light intensity are
presented as 2D color plots in Fig. 2(a-e). In each case, the temporal window
showed in the plots (horizontal axis) is restricted to a fraction
($\sim$1\text{\,}\mathrm{n}\mathrm{s}$$) of the actual recorded length
($>$250\text{\,}\mathrm{n}\mathrm{s}$$) for clarity. For $\Delta
f=$-7.48\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$$ (Fig.2(a)), the ratio
$f_{c}/\Delta f$ calculated from the initial condition is $\sim 1.6$ which
means that only 1st order MI can arise. The observed spatio-temporal dynamics
is reminiscent of the Fermi-Pasta-Ulam-Tsingou recurrences that are usually
observed within this parameter range [22, 24]. Self-compression of the
modulated field leads to the generation of a high contrast train of pulses
(see Fig. 2(f)) followed by a return close to the initial state. This dynamics
is repeated quasi-periodically along propagation. Note that in our case, the
spectrum of the initial condition is asymetric, thus the quasi recurrences are
neither in-phase, nor out-of-phase [5] but rather “continuously” drift towards
long times.

For $\Delta
f=$-4.10\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$\leavevmode\nobreak\
\text{and}\leavevmode\nobreak\ $-2.73\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$$
(Fig. 2(b, c) respectively), the initial harmonic modulation self-compresses
towards a coherent pulse train but each of the pulses later experiences a
coherent splitting dynamics which is the signature of higher order MI [7, 8].
When $\Delta f=$-4.10\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$$, a clear split
in two is observed after $120\text{\,}\mathrm{k}\mathrm{m}$ of propagation
(Fig. 2(g)) which precedes a complex yet still remarkably coherent dynamical
evolution of the pulse train. For $\Delta
f=$-2.73\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$$, the splitting cascade
continues further as the initial modulation turns into regular triplets of
pulses after $224\text{\,}\mathrm{k}\mathrm{m}$ (Fig. 2(h)). Importantly, we
note that the initial ratios $f_{c}/\Delta f$ for these 2 recordings are 2.8
and 4.3 which means that at least 2nd and 4th order MI should occur
respectively in the conservative case. In practice, we indeed observe a
dynamics compatible with 2nd order MI for $\Delta
f=$-4.10\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$$ but only 3rd order MI for
$\Delta f=$-2.73\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$$ as a consequence of
dissipation. Indeed, the exponential decay of the power reduces the cutoff
frequency of the MI gain which decreases the ratio $f_{c}/\Delta f$ during
propagation such that the initial value of this ratio necessarily surestimates
the actual order of MI that can be observed when there is dissipation. It is
important to point out that evolutions very similar to the regular splitting
sequence observed in our experiments appear during the propagation of higher-
order solitons and more generally in the evolution of any arbitrary shaped
pulse of sufficiently large peak power [3, 8, 25, 26, 27]. This suggests that
the spatio-temporal dynamics induced by higher-order MI at early stage is
almost exclusively dictated by the local gradient (hump) of the initial
optical power, i.e. there is no coupling between adjacent modulation periods.

Figure 2: (a-e) Experimental recordings of the spatio-temporal evolution of
initially modulated wavefields for different frequency detunings. (f-j)
Temporal traces extracted from (a-e) at the propagation distances indicated by
the arrows. The traces in light gray on the background are taken after one
circulation ($8\text{\,}\mathrm{k}\mathrm{m}$). Experiments reported in (a-e)
are associated to ratios $f_{c}/\Delta f=1.6,\leavevmode\nobreak\
2.8,\leavevmode\nobreak\ 4.3,\leavevmode\nobreak\ 6.3$ and $16.7$
respectively, $f_{c}$ being estimated at the initial condition (for
$P_{p}\sim$24\text{\,}\mathrm{m}\mathrm{W}$$).

When the frequency detuning is further decreased, the coherence of the spatio-
temporal dynamics is partially lost as shown in Fig. 2(d) due to an emergent
competion with noise-induced MI. Indeed, we observed that each hump of the
initial condition exhibits a different spatio-temporal dynamics and often does
not experience compression to a single pulse followed by a regular splitting
cascade. As an example, the hump on the left part of Fig. 2(d) shows a complex
dynamics that features a degree of stochasticity higher than its neighbours.
Noticeably though, the field after $328\text{\,}\mathrm{k}\mathrm{m}$ of
propagation forms almost perfectly regular quartets of pulses (Fig. 2(i)).
Numerical simulations allows to impute this to the effective dissipation of
the system. For a frequency detuning as low as
$-0.70\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$, the spatio-temporal dynamics
is completely driven by spontaneous MI as depicted in Fig. 2(e). Indeed, the
frequency of the initial modulation is nearly 10 times lower than the
frequency of maximum MI gain which means that spectral noise close to the
maximum gain experiences a growth rate significantly larger than that of the
seed field. Each hump of the initial modulation evolves as a slowly modulated
wavefield that breaks up into a train of almost regular pulses (see Fig. 2(j))
that later evolves stochasticaly. Interestingly, the maximum contrast of the
pulse train is obtained earlier at the times where the initial modulation is
maximum. A related behaviour has been described numerically and analytically
in Ref. [28] using the Akhmediev breather formalism but the present
observation shows that it is a more general phenomenon because it is also
relevant to the case of spontaneous MI. Note that for $\Delta f>0$, we get
symetrical observations to those reported in Fig. 2. Also, we have performed
similar experiments with greater pump-to-seed power ratio which resulted in
qualitatively identical observations.

The gradual loss of coherence that is observed in our experiments originates
from the competition between seeded and spontaneous MI that is mediated, for a
fixed level of spectral noise, by the frequency detuning $\Delta f$. Note that
it has been shown numerically that a similar interplay could disrupt the FPUT
recurrences induced by 1st order MI [29]. In order to get a quantitative
insight into this transition we focus on the evolution of a statistical
indicator which is the second-order moment of the power
$\kappa_{4}(z)=\left\langle P(z,t)^{2}\right\rangle/\left\langle
P(z,t)\right\rangle^{2}$, where $\left\langle\cdot\right\rangle$ denotes
average over time $t$. Larger values of $\kappa_{4}(z)$ traduce the presence,
at given distance $z$, of an increased number of localised structures having
high peak power. In the case of cw spontaneous MI, the value of
$\kappa_{4}(z)$ varies from 1 at initial state to 2 at long propagation
distance and features damped oscillations with a first overshoot that
pinpoints the statistical position of the first maximum compression of the
pulse train generated by MI [21]. Figure 3 shows the longitudinal evolution of
$\kappa_{4}(z)$ calculated for the 5 values of $\Delta f$ presented in Fig. 2
as blue shaded areas. For comparison, the evolution of $\kappa_{4}(z)$
computed from numerical simulations of the NLS equation without inclusion of
spectral noise is superimposed as dashed gray lines. The initial field for the
simulation is $\Psi(0,t)=[\sqrt{P_{p}}+\sqrt{P_{s}}\exp(i2\pi\Delta f\times
t)]$ where $P_{p}$ and $P_{s}$ are the initial pump and seed powers
respectively. Figure 3 shows that our statistical treatment provides clear
identification of the influence of spontaneous MI over the dynamics of seeded
MI. In the case of 1st order MI (Fig. 3(a)), the evolution of $\kappa_{4}$
presents 4 periods traducing the 4 quasi-recurrences observed in Fig. 2(a). In
Figs. 3(b-c) relative to $\Delta
f=$-4.10\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$$ and
$-2.73\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$, the evolution of $\kappa_{4}$
features a first strong peak (first maximum of compression) followed by a
second one (first coherent splitting) and even a weak third one (second
coherent splitting) in Fig. 3(c). This is in adequation with our
identification of 2nd and 3rd order MI dynamics in the spatio-temporal
diagrams of Fig. 2(b-c). Note that the apparent “fluctuations” of $\kappa_{4}$
observed in Fig. 3(b) after $250\text{\,}\mathrm{k}\mathrm{m}$ actually still
traduce a coherent dynamics (see Fig. 2(b)) but altered by linear dissipation.
The very good agreement between experiments and numerical simuations without
noise in these first 3 cases indicates that the dynamics is almost uniquely
governed by seeded MI. Consistently with the results presented previously,
spontaneous MI dramaticaly alters the coherent dynamics of the wavefield when
$|\Delta f|<$2\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$$. This translates in
the longitudinal evolution of $\kappa_{4}$ shown in Fig. 3(d) by the fact that
the large oscillations in the first $300\text{\,}\mathrm{k}\mathrm{m}$ of
propagation observed in the numerical simulation without noise are almost
completely smoothen out in experiments. For even smaller frequency detuning
(Fig. 3(e)), coherent oscillations of $\kappa_{4}$ vanish and we observe a
weak overshoot at $160\text{\,}\mathrm{k}\mathrm{m}$ that is reminiscent of
spontaneous MI. A complete picture of the evolution of $\kappa_{4}(z)$ when
scanning $\Delta f$ is presented as a 2D-color plot in the Supplemental
document along with corresponding numerical simulations to further illustrate
the progressive competition between seeded and spontaneous MI.

Figure 3: Evolution of the second order moment $\kappa_{4}$ calculated from
the experimental recordings illlustrated in Fig. 2. Evolution of $\kappa_{4}$
computed from numerical simulations without inclusion of noise is superimposed
as dashed grey lines. $P_{p}=$24\text{\,}\mathrm{m}\mathrm{W}$$,
$P_{s}=$3\text{\,}\mathrm{m}\mathrm{W}$$,
$\alpha_{\text{eff}}=$3.36\text{\times}{10}^{-6}\text{\,}\mathrm{m}^{-1}$$.

Finally, we analyse through numerical simulations the influence of realistic
dissipation and noise on the spatio-temporal evolution of the wavefield when
higher-order MI is involved. To do so, we focus on the case $\Delta
f=$-1.85\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$$ illustrated in Figs. 2(d)
and 3(d) as it particularly highlights the interplay between seeded and
spontaneous MI. Figure 4(a) shows the result of a simulation of the integrable
NLS equation (i.e. with $\alpha_{\text{eff}}=0$) and no added noise. The
longitudinal evolution up to $300\text{\,}\mathrm{k}\mathrm{m}$ shows a
regular splitting cascade up to 5th-order before the spatio-temporal dynamics
becomes more complex while preserving its temporal coherence. Note that the
observed temporal asymmetry is solely due to the single-sideband excitation.
The addition of linear dissipation to the simulation changes quantitatively
the observed result (see Fig. 4(b)) since the coherent splitting cascade is
stopped at 4th-order. Additional inclusion of realistic noise to the initial
condition results in the spatio-temporal dynamics illustrated in Fig. 4(c)
which agrees very well with our experimental observation of Fig. 2(d). The
spectral noise added to the initial state triggers spontaneous MI which
partially disrupt the coherence of the higher-order MI.

Figure 4: Numerical simulations of Eq. (1) of the experiment presented in Fig.
2(d). (a) Without dissipation nor initial noise, (b) with linear dissipation,
(c) with linear dissipation and inclusion of initial noise (realistic
simulation).

In conclusion, we have used a recirculating optical fiber loop system to
observe in real-time the spatio-temporal evolution of quasi-continuous
wavefields experiencing seeded higher-order modulation instability in a
recirculating optical fiber loop system. Fine scanning of the frequency
detuning between a pump and seed laser enabled clear observation of the
occurence of different orders of modulation instability which result in a
characteristic coherent pulse splitting cascade. At low frequency detuning,
our experiments reveal the coexistence of seeded and spontaneous MI, of which
we analyse the interplay using statistical tools.

Funding. Agence Nationale de la Recherche (StormWave ANR-21-CE30-0009 and
SOGOOD ANR-21-CE30-0061 projects, LABEX CEMPI ANR-11-LABX-0007, I-SITE ULNE
ANR-16-IDEX-0004); Ministry of Higher Education and Research, Hauts de France
council and ERDF through the CPER P4S.

Disclosures. The authors declare no conflicts of interest.

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