# High-precision Electric Dipole Polarizabilities of the Clock States in 133Cs

a,bA. Chakraborty  0000-0001-6255-4584<EMAIL_ADDRESS>aB. K. Sahoo
0000-0003-4397-7965<EMAIL_ADDRESS>aAtomic, Molecular and Optical Physics
Division, Physical Research Laboratory, Navrangpura, Ahmedabad 380009, India
bIndian Institute of Technology Gandhinagar, Palaj, Gandhinagar 382355, India

###### Abstract

We have calculated static and dynamic electric dipole (E1) polarizabilities
($\alpha_{F}$) of the hyperfine levels of the clock transition precisely in
133Cs. The scalar, vector, and tensor components of $\alpha_{F}$ are estimated
by expressing as sum of valence, core, core-core, core-valence, and valence-
core contributions that are arising from the virtual and core intermediate
states. The dominant valence contributions are estimated by combining a large
number of matrix elements of the E1 and magnetic dipole hyperfine interaction
operators from the relativistic coupled-cluster method and measurements. For
an insightful understanding of their accurate determination, we explicitly
give intermediate contributions in different forms to the above quantities.
Very good agreement of the static values for the scalar and tensor components
with their experimental results suggest that our estimated dynamic
$\alpha_{F}$ values can be used reliably to estimate the Stark shifts while
conducting high-precision measurements at the respective laser frequency using
the clock states of 133Cs.

##  I Introduction

Precise estimations of electric dipole polarizabilities ($\alpha_{d}$) are
useful for various high-precision experiments including atom trapping, atomic
clocks, and quantum computers [1, 2, 3, 4, 5]. Among all atoms in the periodic
table, alkali atoms are treated to be very special as they are being
considered in many laboratories to carry out high-precision experiments [6,
7]. Atomic clocks based on the Rb and Cs atoms are frequently used for both
laboratory and space applications [8]. It is also a well known fact that 137Cs
atomic clock is being used as the primary time and frequency standards [9,
10]. In this clock, microwave transition frequency between the hyperfine
levels $F=3$ and $F=4$ of the ground state of 133Cs is used. Since accuracy of
a 133Cs microwave clock is limited by large systematic effects [11, 12],
precise determination of electric dipole (E1) polarizabilities for estimating
the Stark effects of the clock states are quite useful.

The other promising application of the transition between the $F=3$ and $F=4$
ground state hyperfine levels ($|FM_{F}\rangle$) of 137Cs is to make them as
qubits for quantum computers. To realize reliable quantum control and ensure
high fidelity for these applications in quantum science and technology, it is
imperative to minimize decoherence in the single trapped atoms [13]. When an
atomic qubit is encoded as a superposition of two hyperfine levels within the
ground states of an alkali-metal atom, it encounters imbalanced light shifts
induced by the trapping laser field [14, 15, 16, 17]. Consequently, a thorough
analysis of systematic effects is required to understand the influence of the
trapping laser beam’s wavelength, polarization, and intensity on the energy
levels.

From the point of view of studying parity violation (PV) effects in atomic
systems, 133Cs is also very unique as it is the only atom in which electric
dipole amplitude between the $|FM_{F}\rangle$ levels of the ground and 7S
states due to PV has been measured to sub-one percent accuracy [18]. This has
implication for inferring effects beyond the Standard Model of particle
physics. In fact, measuring PV amplitude of the transition between the $F=3$
and $F=4$ hyperfine levels of the ground state in 133Cs would be of particular
interest for probing spin-dependent PV effect. Such an experiment would also
require precise values of the E1 polarizabilities of the involved hyperfine
levels to estimate the systematic effects.

In this paper, we focus on the accurate determination of E1 polarizabilities
($\alpha_{F,M_{F}}$) of the $|FM_{F}\rangle$ levels of the ground state in
133Cs. The differential shift in the clock transition between these hyperfine
levels due to background blackbody radiation (BBR) has recently sparked
interest to estimate the $\alpha_{F,M_{F}}$ values accurately [12]. Several
research groups have extensively investigated the impact of a static electric
field on the hyperfine levels of the ground state in the 133Cs atom [19, 20,
21, 22, 23, 24, 25]. However, there are discrepancies about 10% among the
calculated results on the differential scalar E1 polarizability values from
various methods. This discrepancy is further compounded by variations observed
in different experimental results [26, 27, 28, 29, 30]. Subsequently, it was
claimed that these inconsistencies could be attributed to the neglected
contributions of intermediate continuum states in certain calculations [24].
Similar discrepancy was also seen between the theoretical and experimental
findings for the tensor component of $\alpha_{F,M_{F}}$ [16]. However, it was
later discovered that there was a sign mistake in the theoretical formulation
[31, 32]. Later Dzuba et al. utilized the time-dependent Hartree-Fock (TDHF)
method (equivalent to random phase approximation (RPA)) in conjunction with
Brueckner orbitals (BO) to estimate the tensor polarizability, incorporating
the corrected formula for the hyperfine levels [33]. Even then, the obtained
TDHF result for the $F=4$ level deviated from the experimental value by
approximately 30% [34]. Such substantial discrepancies in both the scalar and
tensor components of the static $\alpha_{F,M_{F}}$ values in the ground state
of 133Cs demands for further investigations on these quantities.

Figure 1: Goldstone diagrams representing the DHF contributions to the second-
order E1 polarizability of the ground state of 133Cs. Here, double arrows
represent valence orbital (v), single arrows going down mean occupied orbitals
(a), and single arrows going up mean virtual orbitals (p). The E1 operator $D$
is represented by the horizontal line.

We carry out analyses of both the static and dynamic $\alpha_{F,M_{F}}$ values
of the hyperfine levels of the ground state in the 133Cs atom. In particular,
we have determined the dynamic $\alpha_{F,M_{F}}$ values at two wavelengths
($\lambda=2\pi c/\omega$ with the speed of light $c$ and angular frequency
$\omega$), namely 936 nm and 1064 nm, for two specific reasons. The
$\lambda=936$ nm value aligns closely with the magic wavelength for the
6$S_{1/2}$ \- 6$P_{3/2}$ transition, which is widely employed for effective
laser cooling of the 133Cs atoms [35, 36]. However, the available powers of
lasers around 936 nm are limited to a few Watts (W). Conversely, the ytterbium
doped fiber laser at $\lambda=1064$ nm offers more than 50 W of power and is
frequently used in laboratories. First, we verify the accuracy of the static
$\alpha_{F,M_{F}}$ values compared with the available experimental and other
theoretical results. Based on these analyses, accuracy of the dynamic
$\alpha_{F,M_{F}}$ values are gauged.

##  II Theory

A uniform oscillating electric field with angular frequency $\omega$ at a
given time $t$ is given by

$\displaystyle\vec{\cal E}_{L}(\omega,t)=\frac{1}{2}|{\cal
E}_{0}|\vec{\varepsilon}e^{-i\omega t}+\text{c.c.},$ (1)

where $|{\cal E}_{0}|$ is the strength of the field, $\vec{\varepsilon}$ is
the degree of polarization and c.c. means complex conjugate term. Interaction
of $\vec{\cal E}_{L}(\omega,t)$ with an atom can be described by the
interaction Hamiltonian

$\displaystyle H_{int}$ $\displaystyle=$ $\displaystyle-\vec{\cal
E}_{L}(\omega,t)\cdot\vec{D}$ (2) $\displaystyle=$ $\displaystyle-\frac{|{\cal
E}_{0}|}{2}\left[\vec{\varepsilon}\cdot\vec{D}e^{-i\omega
t}+{\vec{\varepsilon}}^{*}\cdot\vec{D}e^{i\omega t}\right],$

where $\vec{D}$ is the E1 operator. Since $H_{int}$ is an odd-parity operator,
the first-order shift to the energy levels of atomic states diminishes and the
leading second-order energy shift in power of $|{\cal E}_{0}|$ in a hyperfine
level $|FM_{F}\rangle$ can be given by

$\displaystyle\Delta E_{\rm{light}}=-\frac{1}{2}\alpha_{F,M_{F}}(\omega){\cal
E}_{L}^{2}(\omega),$ (3)

where $\alpha_{F,M_{F}}(\omega)$ is known as the dynamic E1 polarizability and
it corresponds to the static E1 polarizability when $\omega=0$. It would be
imperative to have knowledge of $\alpha_{F,M_{F}}(\omega)$ to estimate $\Delta
E_{\rm{light}}$ at arbitrary values of $|{\cal E}_{0}|$ and $\omega$.
$\alpha_{F,M_{F}}(\omega)$ can be evaluated as expectation value of an
effective operator

$\displaystyle D_{eff}^{(2)}$ $\displaystyle=$
$\displaystyle\left[{\vec{\varepsilon}}^{*}\cdot\vec{D}R^{+}_{F}\vec{\varepsilon}\cdot\vec{D}+\vec{\varepsilon}\cdot\vec{D}R_{F}^{-}{\vec{\varepsilon}}^{*}\cdot\vec{D}\right],\
\ \ $ (4)

where $R_{F}^{\pm}$ are the resolvent operators, given by

$\displaystyle R_{F}^{\pm}$ $\displaystyle=$
$\displaystyle\sum_{F^{\prime},M_{F^{\prime}}}\frac{|F^{\prime}M_{F^{\prime}}\rangle\langle
F^{\prime}M_{F^{\prime}}|}{E_{F}-E_{F^{\prime}}\pm\omega}.$ (5)

Figure 2: Goldstone diagrams representing the top contribution to the third-
order hyperfine interaction induced E1 polarizability. Each diagram contains a
hyperfine interaction $\bf{T_{J}^{(1)}}$ (shown by curly line) in addition to
two interactions by the E1 operator $D$ (shown by horizontal line).

It is possible to separate polarization vectors from the electronic operators
from Eq. (4) by expressing

$\displaystyle{\vec{\varepsilon}}^{*}\cdot\vec{D}R^{\pm}_{F}\vec{\varepsilon}\cdot\vec{D}=\sum_{L=0,1,2}(-1)^{L}\left({\vec{\varepsilon}}^{*}\otimes\vec{\varepsilon}\right)^{L}\cdot\left(\vec{D}\otimes
R^{\pm}_{F}\vec{D}\right)^{L}.$ (6)

Thus, the effective operator is given by

$\displaystyle D_{eff}^{(2)}$ $\displaystyle=$
$\displaystyle\sum_{L=0,1,2}(-1)^{L}\left({\vec{\varepsilon}}^{*}\otimes\vec{\varepsilon}\right)^{L}\cdot\left[\left(\vec{D}\otimes
R^{+}_{F}\vec{D}\right)^{L}\right.$ (7)
$\displaystyle\left.+(-1)^{L}\left(\vec{D}\otimes
R^{-}_{F}\vec{D}\right)^{L}\right].$

using which, we get

$\displaystyle\alpha_{F,M_{F}}$ $\displaystyle=$ $\displaystyle-\langle
FM_{F}|D_{eff}^{(2)}|FM_{F}\rangle$ (8) $\displaystyle=$
$\displaystyle-\sum_{L=0,1,2}\sum_{Q=-L}^{L}(-1)^{L-Q}\left({\vec{\varepsilon}}^{*}\otimes\vec{\varepsilon}\right)^{L}_{Q}$
$\displaystyle\times\langle FM_{F}|\left(\vec{D}\otimes
R^{+}_{F}\vec{D}\right)^{L}_{Q}$ $\displaystyle+(-1)^{L}\langle
FM_{F}|\left(\vec{D}\otimes R^{-}_{F}\vec{D}\right)^{L}_{Q}|FM_{F}\rangle.$

Figure 3: Goldstone diagrams representing the center part of the third-order
hyperfine interaction induced E1 polarizability. All notations are same with
the previous two figures.

Using the polarization dependent factors, we can rewrite the aforementioned
expression as

$\displaystyle\alpha_{F,M_{F}}$ $\displaystyle=$
$\displaystyle\alpha_{F}^{S}+{\cal
A}\frac{M_{F}}{2F}\cos{\theta_{k}}\alpha_{F}^{A}$ (9)
$\displaystyle+\frac{3M_{F}^{2}-F(F+1)}{F(2F-1)}\frac{3\cos^{2}{\theta_{p}}-1}{2}\alpha_{F}^{T},$

where $\theta_{k}$ is the angle between the wave vector and quantization axis,
$\theta_{p}$ is the polarization angle and ${\cal A}$ denotes degree of
polarization. Again, $\alpha_{F}^{S}$, $\alpha_{F}^{A}$, and $\alpha_{F}^{T}$
are known as the scalar, axial-vector, and tensor components of
$\alpha_{F,M_{F}}$, which are $M_{F}$ independent and are given by

$\displaystyle\alpha_{F}^{S}(\omega)$ $\displaystyle=$
$\displaystyle-\frac{1}{3(2F+1)}\sum_{F^{\prime}}|\langle F||{\bf
D}||F^{\prime}\rangle|^{2}$ (10)
$\displaystyle\times\left[\frac{1}{E_{F}-E_{F^{\prime}}+\omega}+\frac{1}{E_{F}-E_{F^{\prime}}-\omega}\right],\
\ $ $\displaystyle\alpha_{F}^{A}(\omega)$ $\displaystyle=$
$\displaystyle-\sqrt{\frac{6F}{(F+1)(2F+1)}}\sum_{F^{\prime}}(-1)^{F+F^{\prime}+1}$
(14) $\displaystyle\times\left\\{\begin{array}[]{ccc}F&1&F\\\
1&F^{\prime}&1\end{array}\right\\}|\langle F||{\bf D}||F^{\prime}\rangle|^{2}$
$\displaystyle\times\left[\frac{1}{E_{F}-E_{F^{\prime}}+\omega}-\frac{1}{E_{F}-E_{F^{\prime}}-\omega}\right],$

and

$\displaystyle\alpha_{F}^{T}(\omega)$ $\displaystyle=$ $\displaystyle
2\sqrt{\frac{5F(2F-1)}{6(F+1)(2F+3)(2F+1)}}$ (18)
$\displaystyle\times(-1)^{F+F^{\prime}+1}\left\\{\begin{array}[]{ccc}F&2&F\\\
1&F^{\prime}&1\end{array}\right\\}|\langle F||{\bf D}||F^{\prime}\rangle|^{2}$
$\displaystyle\times\left[\frac{1}{E_{F}-E_{F^{\prime}}+\omega}+\frac{1}{E_{F}-E_{F^{\prime}}-\omega}\right].$

Figure 4: Goldstone diagrams representing the normalization part of the third-order hyperfine interaction induced E1 polarizability. This has similarity with the diagrams representing the second-order E1 polarizability. Table 1: Calculated values of the second-order static and dynamic E1 polarizabilities (in a.u.) of the ground state of the Cs atom. Method |  $\alpha_{6S}^{S}$ values |  |  $\alpha_{6S}^{A}$ values
---|---|---|---
|  $\lambda=\infty$ |  $\lambda=936$ nm |  $\lambda=1064$ nm |  |  $\lambda=936$ nm |  $\lambda=1064$ nm
This work
DHF |  $662.6$ |  $-2303.2$ |  $7945.1$ |  |  $-459.7$ |  $20772.2$
RCCSD |  $404.8$ |  $2684.0$ |  $1138.7$ |  |  $-1300.8$ |  $-196.8$
RCCSDT |  $400.0$ |  $3094.3$ |  $1164.4$ |  |  $-1819.3$ |  $-206.3$
Final |  ${\bf 401.0(6)}$ |  ${\bf 3022.1(40)}$ |  ${\bf 1170.8(16)}$ |  |  ${\bf-1599.5(59)}$ |  ${\bf-201.8(18)}$
Others
Theory [25] |  $400.80(97)$ |  |  |  |  |
Theory [50] |  $399.8$ |  |  |  |  |
Theory [51] |  $403.9$ |  |  |  |  |
Theory [52] |  $399.9(1.9)$ |  |  |  |  |
Experiment [53] |  401.00(6) |  |  |  |  |

It is strenuous to deal with the wave functions in the hyperfine coordinate
system to evaluate the above quantities. To address this, we can express the
$|FM_{F}\rangle$ levels with a good approximation considering up to the first-
order perturbation as

$\displaystyle|FM_{F}\rangle$ $\displaystyle=$
$\displaystyle|IM_{I};JM_{J}\rangle+\sum_{J^{\prime},M_{J^{\prime}}}|IM_{I};J^{\prime}M_{J^{\prime}}\rangle$
(19) $\displaystyle\times\frac{\langle
IM_{I};J^{\prime}M_{J^{\prime}}|H_{hf}|IM_{I};JM_{J}\rangle}{E_{J}-E_{J^{\prime}}},$

where $I$ is the nuclear spin with azimuthal component $M_{I}$ and $J$ is the
total angular momentum of the atomic state with azimuthal component $M_{J}$.
In the above expression, $H_{hf}$ denotes the scalar hyperfine interaction
Hamiltonian, which can be defined as

$\displaystyle H_{hf}=\sum_{k}T_{J}^{(k)}\cdot T_{I}^{(k)},$ (20)

where $T_{J}^{(k)}$ and $T_{I}^{(k)}$ are defined as the electronic and
nuclear components, respectively, of $H_{hf}$ with rank $k$ of the multipole
expansion with $k=1,3,5\cdots$ denoting contributions from the magnetic
multipoles while $k=2,4,6\cdots$ give contributions from the electric
multipoles. For the present interest, we consider only the dominant $k=1$ term
in the calculation corresponding to magnetic dipole (M1) hyperfine interaction
as contributions from the other multipoles to these quantities are negligibly
small [32, 33]. The $\langle
IM_{I};J^{\prime}M_{J^{\prime}}|H_{hf}|IM_{I};JM_{J}\rangle$ matrix element
can, then, be evaluated using the relation

$\displaystyle\langle IM_{I};J^{\prime}M_{J^{\prime}}|T_{J}^{(1)}\cdot
T_{I}^{(1)}|IM_{I};JM_{J}\rangle=(-1)^{I+J+F}$
$\displaystyle\times\begin{Bmatrix}J^{\prime}&J&1\cr
I&I&F\cr\end{Bmatrix}\langle J^{\prime}||{\bf T_{J}^{(1)}}||J\rangle\langle
I||{\bf T_{I}^{(1)}}||I\rangle,$ (21)

in which the nuclear coordinate part is converted to a factor as

$\displaystyle\langle I||{\bf
T_{I}^{(1)}}||I\rangle=\sqrt{I(I+1)(2I+1)}g_{I}\mu_{N},$ (22)

with $g_{I}=\mu_{I}/I$ for the M1 moment $\mu_{I}$ and nuclear Bohr magnetron
$\mu_{N}$.

Table 2: Magnetic dipole hyperfine interaction induced E1 polarizabilities (in $10^{-10}$ Hz/(V/m)2) of the hyperfine levels of the ground state of 133Cs at various wavelengths ($\lambda$). The unit Hz/(V/m)2 can be converted into a.u. by multiplying 0.401878046 $\times 10^{8}$.  |  | $F=3$ |  | $F=4$
---|---|---|---|---
Quantity |  Method |  $\lambda=\infty$ |  $\lambda=936$ nm |  $\lambda=1064$ nm |  |  $\lambda=\infty$ |  $\lambda=936$ nm |  $\lambda=1064$ nm
$\alpha_{F}^{S(2,1)}$ |  DHF |  $-3.1420$ |  $-49.5027$ |  $-2381.2965$ |  |  $2.4423$ |  $38.5007$ |  $1852.1969$
|  RCCSD |  $-2.5706$ |  $-153.5968$ |  $-26.5174$ |  |  $1.9993$ |  $119.4880$ |  $8.3956$
|  RCCSDT |  $-2.5586$ |  $-225.2741$ |  $-25.3313$ |  |  $1.9898$ |  $175.2118$ |  $19.6881$
|  Final |  ${\bf-2.559(11)}$ |  ${\bf-201.1(17)}$ |  ${\bf-25.3(13)}$ |  |  ${\bf 1.990(10)}$ |  ${\bf 156.4(14)}$ |  ${\bf 19.7(10)}$
|  TDHF$+$BO [33] |  $-2.5419$ |  |  |  |  $1.9770$ |  |
|  RCICP [25] |  $-34.248(7)$ |  |  |  |  $-29.598(7)$ |  |
$\alpha_{F}^{A(2,1)}$ |  DHF |  $0.0$ |  $8.6958$ |  $561.5658$ |  |  $0.0$ |  $9.0179$ |  $582.5113$
|  RCCSD |  $0.0$ |  $-132.2379$ |  $-9.0495$ |  |  $0.0$ |  $-137.1366$ |  $-9.2136$
|  RCCSDT |  $0.0$ |  $-238.6758$ |  $-11.0932$ |  |  $0.0$ |  $-247.5169$ |  $-11.5043$
|  Final |  $0.0$ |  ${\bf-185.59(51)}$ |  ${\bf-9.70(7)}$ |  |  $0.0$ |  ${\bf-192.47(53)}$ |  ${\bf-10.06(7)}$
$\alpha_{F}^{T(2,1)}$ |  DHF |  $0.0344$ |  $0.4310$ |  $25.8040$ |  |  $-0.0639$ |  $-0.8044$ |  $-48.1693$
|  RCCSD |  $0.0183$ |  $6.0153$ |  $0.4561$ |  |  $-0.0339$ |  $-11.2287$ |  $-0.8888$
|  RCCSDT |  $0.0188$ |  $10.4966$ |  $0.5508$ |  |  $-0.0350$ |  $-19.5937$ |  $-1.0279$
|  Final |  ${\bf 0.0185(8)}$ |  ${\bf 8.482(16)}$ |  ${\bf 0.5084(21)}$ |  |  ${\bf-0.0342(15)}$ |  ${\bf-15.834(30)}$ |  ${\bf-0.9487(39)}$
|  TDHF$+$BO [33] |  $0.0141$ |  |  |  |  $-0.0262$ |  |
|  RCICP [25] |  $0.03051(6)$ |  |  |  |  $-0.05703(11)$ |  |
|  Semi-empirical [31] |  |  |  |  |  $-0.0372(25)$ |  |
|  Experiment [34] |  |  |  |  | $-0.0334(2)_{stat}(25)_{syst}$

After substituting all the relations, we can express $\alpha_{F}^{S}$,
$\alpha_{F}^{A}$ and $\alpha_{F}^{T}$ components as

$\displaystyle\alpha_{F}^{S}$ $\displaystyle=$
$\displaystyle\alpha_{F}^{S(2,0)}+\alpha_{F}^{S(2,1)},$ (23)
$\displaystyle\alpha_{F}^{A}$ $\displaystyle=$
$\displaystyle\alpha_{F}^{A(2,0)}+\alpha_{F}^{A(2,1)},$ (24)

and

$\displaystyle\alpha_{F}^{T}$ $\displaystyle=$
$\displaystyle\alpha_{F}^{T(2,0)}+\alpha_{F}^{T(2,1)},$ (25)

where $\alpha_{F}^{S/A/T(m,n)}$ means the components are including $m$-orders
of E1 interactions and $n$-orders of M1 interactions, respectively. The
hyperfine interaction independent components can be evaluated conveniently now
by using the relations

$\displaystyle\alpha_{F}^{S(2,0)}(\omega)$ $\displaystyle=$
$\displaystyle-\frac{1}{3(2J+1)}\sum_{J^{\prime}}|\langle J||{\bf
D}||J^{\prime}\rangle|^{2}$ (26)
$\displaystyle\times\left[\frac{1}{E_{J}-E_{J^{\prime}}+\omega}+\frac{1}{E_{J}-E_{J^{\prime}}-\omega}\right]$
$\displaystyle\equiv$ $\displaystyle\alpha_{J}^{S}(\omega),$
$\displaystyle\alpha_{F}^{A(2,0)}(\omega)$ $\displaystyle=$
$\displaystyle-\sqrt{\frac{6F(2F+1)}{(F+1)}}\left\\{\begin{array}[]{ccc}J&F&I\\\
F&J&1\end{array}\right\\}$ (35)
$\displaystyle\times\sum_{J^{\prime}}(-1)^{F+J^{\prime}+I+2J}\left\\{\begin{array}[]{ccc}1&1&1\\\
J&J&J^{\prime}\end{array}\right\\}$ $\displaystyle\times\left[\frac{|\langle
J||{\bf
D}||J^{\prime}\rangle|^{2}}{E_{J}-E_{J^{\prime}}+\omega}-\frac{|\langle
J||{\bf D}||J^{\prime}\rangle|^{2}}{E_{J}-E_{J^{\prime}}-\omega}\right]$
$\displaystyle=$ $\displaystyle\sqrt{\frac{F(2F+1)(J+1)(2J+1)}{J(F+1)}}$
$\displaystyle\times(-1)^{I+J+F+1}\left\\{\begin{array}[]{ccc}J&F&I\\\
F&J&1\end{array}\right\\}\alpha_{J}^{A}(\omega),\ \ \ $

and

$\displaystyle\alpha_{F}^{T(2,0)}(\omega)$ $\displaystyle=$
$\displaystyle-\sqrt{\frac{20F(2F-1)(2F+1)}{6(F+1)(2F+3)}}\left\\{\begin{array}[]{ccc}J&F&I\\\
F&J&2\end{array}\right\\}$ (45)
$\displaystyle\times\sum_{J^{\prime}}(-1)^{I+F+J^{\prime}+2J}\left\\{\begin{array}[]{ccc}1&1&2\\\
J&J&J^{\prime}\end{array}\right\\}$ $\displaystyle\times\left[\frac{|\langle
J||{\bf
D}||J^{\prime}\rangle|^{2}}{E_{J}-E_{J^{\prime}}+\omega}+\frac{|\langle
J||{\bf D}||J^{\prime}\rangle|^{2}}{E_{J}-E_{J^{\prime}}-\omega}\right]$
$\displaystyle=$
$\displaystyle-\sqrt{\frac{(J+1)(2J+3)(2J+1)F(2F-1)}{J(2J-1)(F+1)(2F+3)(2F+1)}}$
$\displaystyle\times(2F+1)(-1)^{I+J+F+1}\left\\{\begin{array}[]{ccc}J&F&I\\\
F&J&2\end{array}\right\\}$ $\displaystyle\times\ \ \ \alpha_{J}^{T}(\omega),$

where $\alpha_{J}^{S}$, $\alpha_{J}^{A}$ and $\alpha_{J}^{T}$ are nothing but
the components of atomic state E1 polarizabilities whose evaluations depend on
the electronic wave functions and energies only. It can be followed from the
selection rules that $\alpha_{J}^{T}$ will not contribute to the states with
$J<3/2$.

Proceeding in the similar manner, we can express [24, 37, 38]

$\displaystyle\alpha^{{\cal K}(2,1)}_{F}(\omega)=W_{F}^{\cal
K}\left[2T_{F}^{\cal K}(\omega)+C_{F}^{\cal K}(\omega)+R_{F}^{\cal
K}(\omega)\right],$ (46)

where the symbol ${\cal K}$ denotes scalar, axial-vector, and tensor
components for the integer values $K=0$, 1 and 2, respectively, as used below.
Here, each component is divided into contributions from three different terms
defined as top ($T_{F}^{\cal K}$), center ($C_{F}^{\cal K}$), and residual (or
normalization) ($R_{F}^{\cal K}$) that are given by

$\displaystyle T_{F}^{\cal K}(\omega)$ $\displaystyle=$
$\displaystyle\sqrt{(2K+1)I(I+1)(2I+1)}g_{I}\mu_{N}$ $\displaystyle\times$
$\displaystyle\sum_{J^{\prime},J^{\prime\prime}}\Bigg{\\{}\begin{array}[]{c c
c}I&I&1\\\
J&J^{\prime\prime}&F\end{array}\Bigg{\\}}\Bigg{\\{}\begin{array}[]{c c
c}K&J^{\prime\prime}&J\\\ I&F&F\end{array}\Bigg{\\}}$ (51)
$\displaystyle\times$ $\displaystyle\Bigg{\\{}\begin{array}[]{c c
c}K&J^{\prime\prime}&J\\\ J^{\prime}&1&1\end{array}\Bigg{\\}}$ (54)
$\displaystyle\times$ $\displaystyle(-1)^{J+J"}\frac{\langle J||{\bf
T}_{J}^{(1)}||J^{\prime\prime}\rangle\langle J^{\prime\prime}||{\bf
D}||J^{\prime}\rangle\langle J^{\prime}||{\bf
D}||J\rangle}{(E_{J}-E_{J^{\prime\prime}})}$ $\displaystyle\times$
$\displaystyle\Bigg{[}\frac{1}{(E_{J}-E_{J^{\prime}}+\omega)}+\frac{(-1)^{K}}{(E_{J}-E_{J^{\prime}}-\omega)}\Bigg{]},$
(55) $\displaystyle C_{F}^{\cal K}(\omega)$ $\displaystyle=$
$\displaystyle\sqrt{(2K+1)I(I+1)(2I+1)}g_{I}\mu_{N}$ $\displaystyle\times$
$\displaystyle\sum_{J^{\prime},J^{\prime\prime}}\sum_{L}\Biggl{\\{}\begin{array}[]{c
c c}F&K&F\\\ J&1&J^{\prime\prime}\\\
I&1&L\end{array}\Biggr{\\}}\Bigg{\\{}\begin{array}[]{c c c}I&J&F\\\
1&J^{\prime}&J^{\prime\prime}\\\ I&1&L\end{array}\Bigg{\\}}$ (62)
$\displaystyle\times$ $\displaystyle(-1)^{I+K-F+J}$ $\displaystyle\times$
$\displaystyle\langle J||{\bf D}||J^{\prime\prime}\rangle\langle
J^{\prime\prime}||{\bf T}_{J}^{(1)}||J^{\prime}\rangle\langle J^{\prime}||{\bf
D}||J\rangle$ $\displaystyle\times$
$\displaystyle\Bigg{[}\frac{1}{(E_{J}-E_{J^{\prime}}+\omega)(E_{J}-E_{J^{\prime\prime}}+\omega)}$
$\displaystyle+$
$\displaystyle\frac{(-1)^{K}}{(E_{J}-E_{J^{\prime}}-\omega)(E_{J}-E_{J^{\prime\prime}}-\omega)}\Bigg{]},$
(63)

and

$\displaystyle R_{F}^{\cal K}(\omega)$ $\displaystyle=$
$\displaystyle\sqrt{(2K+1)I(I+1)(2I+1)}g_{I}\mu_{N}$ $\displaystyle\times$
$\displaystyle\sum_{J^{\prime}}\Bigg{\\{}\begin{array}[]{c c c}I&I&1\\\
J&J&F\end{array}\Bigg{\\}}\Bigg{\\{}\begin{array}[]{c c c}K&J&J\\\
I&F&F\end{array}\Bigg{\\}}\Bigg{\\{}\begin{array}[]{c c c}K&J&J\\\
J^{\prime}&1&1\end{array}\Bigg{\\}}$ (70) $\displaystyle\times$
$\displaystyle(-1)^{(J+J^{\prime}+1)}\langle J||{\bf
T}_{J}^{(1)}||J\rangle|\langle J||{\bf D}||J^{\prime}\rangle|^{2}$
$\displaystyle\times$
$\displaystyle\left[\frac{1}{(E_{J}-E_{J^{\prime}}+\omega)^{2}}+\frac{(-1)^{K}}{(E_{J}-E_{J^{\prime}}-\omega)^{2}}\right].\
\ \ \ $ (71)

Also, the pre-angular factors are given by

$\displaystyle W_{F}^{S}$ $\displaystyle=$
$\displaystyle\sqrt{\frac{(2F+1)}{3}},$ (72) $\displaystyle W_{F}^{A}$
$\displaystyle=$ $\displaystyle-\sqrt{\frac{2F(2F+1)}{(F+1)}},$ (73)

and

$\displaystyle W_{F}^{T}$ $\displaystyle=$
$\displaystyle-\sqrt{\frac{2F(2F-1)(2F+1)}{3(F+1)(2F+3)}}.$ (74)

##  III Approaches for evaluation

As can be inferred from the above discussion, we need a large set of matrix
elements of the $D$ and $T_{J}^{(1)}$ operators for precise estimate of the
$\alpha_{F}$ values in 133Cs. Since wave functions of the atomic states of
133Cs cannot be solved exactly, we can determine these matrix elements using a
mean-field approximation. We use the Dirac-Hartree-Fock (DHF) approach to
obtain mean-field wave functions of the Dirac-Coulomb (DC) Hamiltonian, which
in atomic unit (a.u.) is given by

$\displaystyle H_{DC}$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{N_{e}}\left[c{\vec{\alpha}}_{D}\cdot{\vec{p}}_{i}+(\beta-1)c^{2}+V_{n}(r_{i})\right]+\sum_{i>j}\frac{1}{r_{ij}},$

where $N_{e}$ is the number of electrons in the atom, ${\vec{\alpha}}_{D}$ and
$\beta$ are the Dirac matrices, $V_{n}(r)$ is the nuclear potential, and
$r_{ij}$ is the inter-electronic distances between electrons located at
$r_{i}$ and $r_{j}$. We have also included corrections due to Breit and lower-
order quantum electrodynamics (QED) to improve accuracy in the calculations.
Within the QED contribution, we have accounted for corrections stemming from
the lowest-order vacuum polarization effect, described through the Uehling
potential and Wichmann-Kroll potential, and self-energy effect described by
the magnetic and electric form-factors [39, 40, 41].

Table 3: The presently calculated the second-order static and dynamic atomic E1 polarizabilities (in a.u.) of the ground state of Cs atom. E1 matrix elements used in the estimation of ‘main’ contributions are given explicitly, where values shown with superscript ‘$a$’ are calculated using the RCCSDT method. Transition | E1 matrix element |  $\alpha_{6S}^{S}$ values |  |  $\alpha_{6S}^{A}$ values
---|---|---|---|---
|  | $\lambda=\infty$ |  $\lambda=936$ nm |  $\lambda=1064$ nm |  |  $\lambda=936$ nm |  $\lambda=1064$ nm
Main |  |  |  |  |  |  |
$6S_{1/2}-6P_{1/2}$ | 4.5067(40)a | 132.93 | 1536.35 | 453.54 |  | $-2936.77$ | $-762.65$
$6S_{1/2}-6P_{3/2}$ | 6.3403(64) [54] | 250.67 | 1467.97 | 699.66 |  | 1336.78 | 560.48
$6S_{1/2}-7P_{1/2}$ | 0.27810(45) [55] | 0.26 | 0.34 | 0.32 |  | $-0.34$ | $-0.28$
$6S_{1/2}-7P_{3/2}$ | 0.57417(57) [55] | 1.10 | 1.44 | 1.35 |  | 0.70 | 0.58
$6S_{1/2}-8P_{1/2}$ | 0.0824(10)a | 0.02 | 0.02 | 0.02 |  | $-0.02$ | $-0.02$
$6S_{1/2}-8P_{3/2}$ | 0.2294(15)a | 0.15 | 0.18 | 0.17 |  | 0.08 | 0.06
$6S_{1/2}-9P_{1/2}$ | 0.0424(15)a | 0.01 | 0.01 | 0.01 |  | $-0.01$ | $\sim 0.0$
$6S_{1/2}-9P_{3/2}$ | 0.1268(11)a | 0.04 | 0.05 | 0.05 |  | 0.02 | 0.02
Total |  | 385.2(6) | 3006.4(40) | 1155.1(16) |  | $-1599.5(59)$ | $-201.8(18)$
Tail |  | 0.20 | 0.14 | 0.14 |  | 0.005 | 0.004
Core-valence |  | $-0.35(5)$ | $-0.35(5)$ | $-0.35(5)$ |  | $-0.01(1)$ | $-0.01(1)$
Core |  | 15.99(10) | 15.9(1) | 15.9(1) |  | $0.0$ | $0.0$

To produce as many bound states having a common core $[5p^{6}]$ but differing
by a valence orbital $v$ in 133Cs as possible, we consider the $V^{N-1}$
potential in the DHF method. In this approach, the DHF wave functions of the
interested states are denoted by

$\displaystyle|\Phi_{v}\rangle=a_{v}^{\dagger}|\Phi_{0}\rangle,$ (75)

where $|\Phi_{0}\rangle$ is the DHF wave function of the closed-core
$[5p^{6}]$. Using these wave functions, we can determine the dominant part of
the $\alpha_{J}^{S}(\omega)$ and $\alpha_{J}^{A}(\omega)$ values of the ground
state of in 133Cs. In Fig. 1, we show Goldstone diagram representations of the
DHF contributions for $\alpha_{J}^{S}(\omega)$ and $\alpha_{J}^{A}(\omega)$.
Since $D$ is a one-body operator, the DHF diagrams include contributions only
from the intermediate states that are represented by single orbital
excitations. Thus, we can classify these diagrams as core, core-valence, and
valence orbital contributions corresponding to Figs. 1(i), (ii), and (iii)
respectively. In order to improve these calculations for precise estimations
of the E1 polarizabilities, it is imperative to include electron correlation
effects arising through other configurations neglected in the DHF method. It
is possible to adopt a linear response approach [42, 43] to include the
electron correlation effects for carrying out ab initio calculations of the
above quantities. However, accuracy of the first-principle results will be
restricted by the uncertainties associated with both the calculated energies
and E1 matrix elements. To minimize uncertainties in the calculations, we
intend to use the experimental energies from the National Institute of Science
and Technology (NIST) database [44] which are known with very high accuracy.
Similarly, we want to use very precise values of the E1 matrix elements either
from the theory or experiments wherever available. First, we attempt to
evaluate these E1 matrix elements using the relativistic coupled-cluster (RCC)
method. Wherever we find the experimental E1 values are available with higher
accuracy than our RCC results, we use the experimental results. However, it
should be noted that the extracted experimental E1 values do not possess
information about their signs, which is essential in the determination of the
hyperfine interaction induced E1 polarizabilities. So, we use our calculated
E1 matrix elements for assigning signs to the precisely known experimental E1
values. Again, contributions from the high-lying continuum orbitals to the
valence contributions are estimated using lower-order methods and quoted as
“tail’ contributions while we list the valence contributions from low-lying
bound states as “main” contributions to distinguish them in the analyses.

In the RCC theory ansatz, wave function of an atomic state with a closed-shell
electronic configuration and a valence orbital can be expressed by [45]

$\displaystyle|\Psi_{v}\rangle$ $\displaystyle=$ $\displaystyle
e^{T}\left\\{1+S_{v}\right\\}|\Phi_{v}\rangle,$ (76)

where $T$ is the RCC operator that accounts for the excitations of core
electrons to virtual orbitals, and $S_{v}$ is the RCC operator that excites
the valence and core orbitals together to virtual orbitals due to the
correlation effects. Amplitudes of the $T$ and $S_{v}$ excitation operators
are obtained by

$\displaystyle\langle\Phi_{0}^{*}|(He^{T})_{c}|\Phi_{0}\rangle=0$ (77)

and

$\displaystyle\langle\Phi_{v}^{*}|[(He^{T})_{c}-E_{v}]S_{v}|\Phi_{v}\rangle=-\langle\Phi_{v}^{*}|(He^{T})_{c}|\Phi_{v}\rangle,$
(78)

where subscript $c$ denotes the connected terms and projected states with
superscript $*$ stand for the excited state Slater determinants with respect
to the respective DHF states. The exact energy of the state is given by

$\displaystyle E_{v}$ $\displaystyle=$
$\displaystyle\langle\Phi_{v}|H_{eff}|\Phi_{v}\rangle=\langle\Phi_{v}|(He^{T})_{c}\left\\{1+S_{v}\right\\}|\Phi_{v}\rangle.\
\ \ $ (79)

We have considered single, double, and triple excitations in the RCC method
(RCCSDT method) by defining

$\displaystyle T=T_{1}+T_{2}+T_{3}$ (80)

and

$\displaystyle S_{v}=S_{1v}+S_{2v}+S_{3v},$ (81)

where subscripts 1, 2, and 3 denote the single, double, and triple excitations
respectively. Since it is challenging to include triple excitations from a
large set of basis functions, we first considered only the single and double
excitations in the RCC method (RCCSD method) for a sufficiently large basis
functions. From the analysis of the results from the RCCSD method, we find out
the most active orbitals that contribute predominantly in 133Cs. Then, we
allow triple excitations only from those selected orbitals in the RCCSDT
method.

After obtaining amplitudes of the RCC operators, matrix element of a physical
operator $O$ between the $|\Psi_{f}\rangle$ and $|\Psi_{i}\rangle$ states is
evaluated by

$\displaystyle\langle O\rangle_{fi}$ $\displaystyle=$
$\displaystyle\frac{\langle\Psi_{f}|O|\Psi_{i}\rangle}{\sqrt{\langle\Psi_{f}|\Psi_{f}\rangle\langle\Psi_{i}|\Psi_{i}\rangle}}$
(82) $\displaystyle=$
$\displaystyle\frac{\langle\Phi_{f}|\\{S_{f}^{\dagger}+1\\}\overline{O}\\{1+S_{i}\\}|\Phi_{i}\rangle}{\langle\Phi_{f}|\\{S_{f}^{\dagger}+1\\}\overline{N}\\{1+S_{i}\\}|\Phi_{i}\rangle},$

where $\overline{O}=e^{T^{\dagger}}Oe^{T}$ and
$\overline{N}=e^{T^{\dagger}}e^{T}$. Both $\overline{O}$ and $\overline{N}$
are the non-terminating series, which are evaluated by adopting iterative
procedures [46, 47, 48].

Table 4: Some of the important matrix elements (in a.u.) of the ${\bf T_{J}^{(1)}}$ operator of 133Cs. Numbers appearing as $a[b]$ mean $a\times 10^{b}$. See the text for details explaining how the experimental values for the off-diagonal matrix elements are inferred. Transition |  RCCSDT method |  Experiment
---|---|---
$6S_{1/2}$-$6S_{1/2}$ |  $5.817[-7]$ |  $5.797[-7]$ [56]
$6S_{1/2}$-$7S_{1/2}$ |  $2.859[-7]$ |  $2.825[-7]$ [56, 57]
$6S_{1/2}$-$8S_{1/2}$ |  $1.795[-7]$ |  $1.790[-7]$ [56, 58]
$6S_{1/2}$-$5D_{3/2}$ |  $-1.674[-8]$ |
$6S_{1/2}$-$6D_{3/2}$ |  $8.770[-9]$ |
$6P_{1/2}$-$6P_{1/2}$ |  $7.341[-8]$ |  $7.364[-8]$ [59]
$6P_{1/2}$-$7P_{1/2}$ |  $4.143[-8]$ |  $4.187[-8]$ [59, 60]
$6P_{1/2}$-$8P_{1/2}$ |  $2.759[-8]$ |  $2.821[-8]$ [59, 61]
$6P_{1/2}$-$7P_{1/2}$ |  $4.143[-8]$ |
$6P_{1/2}$-$9P_{1/2}$ |  $-1.968[-8]$ |
$6P_{1/2}$-$6P_{3/2}$ |  $-4.394[-9]$ |
$6P_{1/2}$-$7P_{3/2}$ |  $-2.572[-9]$ |
$7P_{1/2}$-$7P_{1/2}$ |  $2.371[-8]$ |  $2.381[-8]$ [60]
$7P_{1/2}$-$8P_{1/2}$ |  $1.567[-8]$ |  $1.606[-8]$ [60, 61]
$7P_{1/2}$-$9P_{1/2}$ |  $-11.177[-9]$ |
$7P_{1/2}$-$6P_{3/2}$ |  $-2.402[-9]$ |
$7P_{1/2}$-$7P_{3/2}$ |  $-1.417[-9]$ |
$8P_{1/2}$-$8P_{1/2}$ |  $10.595[-9]$ |  $10.840[-9]$ [61]
$8P_{1/2}$-$9P_{1/2}$ |  $-7.446[-9]$ |
$8P_{1/2}$-$6P_{3/2}$ |  $-1.610[-9]$ |
$8P_{1/2}$-$7P_{3/2}$ |  $-9.460[-10]$ |
$9P_{1/2}$-$9P_{1/2}$ |  $5.313[-9]$ |
$6P_{3/2}$-$6P_{3/2}$ |  $3.874[-8]$ |
$6P_{3/2}$-$7P_{3/2}$ |  $2.214[-8]$ |
$6P_{3/2}$-$8P_{3/2}$ |  $1.500[-8]$ |
$7P_{3/2}$-$7P_{3/2}$ |  $12.648[-9]$ |
Table 5: Breakdown of our calculated $\alpha_{F}^{S(2,1)}$, $\alpha_{F}^{A(2,1)}$ and $\alpha_{F}^{T(2,1)}$ values for the $F=3$ and $F=4$ levels of 133Cs in terms of the valence, valence-core, core-valence, core-core and core contributions. Results are given for both the static and dynamic E1 polarizabilities (in $10^{-10}$ Hz/(V/m)2).  |  | $F=3$ |  | $F=4$ |
---|---|---|---|---|---
Polarizability | Contribution |  | $\lambda=\infty$ |  $\lambda=936$ nm |  $\lambda=1064$ nm |  | $\lambda=\infty$ |  $\lambda=936$ nm |  $\lambda=1064$ nm
$\alpha_{F}^{S(2,1)}$ | Valence |  | $-2.5584$ | $-201.0945$ | $-25.3858$ |  | $1.9904$ | $156.4064$ | $19.7445$
| Valence-Core |  | $-0.0016$ | $-0.0032$ | $0.0601$ |  | $0.0013$ | $0.0025$ | $-0.0467$
| Core-Valence |  | $0.0010$ | $-0.0040$ | $0.0402$ |  | $-0.0008$ | $0.0031$ | $-0.0313$
| Core-Core |  | $-0.0009$ | $-0.0009$ | $-0.0009$ |  | $0.0007$ | $\sim 0.0$ | $0.0007$
| Core |  | $0.0010$ | $0.0010$ | $0.0010$ |  | $-0.0015$ | $-0.0015$ | $-0.0015$
$\alpha_{F}^{A(2,1)}$ | Valence |  | $0.0$ | $-185.6502$ | $-9.6217$ |  | $0.0$ | $-192.5270$ | $-9.9781$
| Valence-Core |  | $0.0$ | $0.0317$ | $-0.0258$ |  | $0.0$ | $0.0329$ | $-0.0268$
| Core-Valence |  | $0.0$ | $0.0265$ | $-0.0548$ |  | $0.0$ | $0.0275$ | $-0.0569$
| Core-Core |  | $0.0$ | $\sim 0.0$ | $\sim 0.0$ |  | $0.0$ | $\sim 0.0$ | $\sim 0.0$
| Core |  | $0.0$ | $\sim 0.0$ | $\sim 0.0$ |  | $0.0$ | $\sim 0.0$ | $\sim 0.0$
$\alpha_{F}^{T(2,1)}$ | Valence |  | $0.0165$ | $8.4872$ | $0.5794$ |  | $-0.0308$ | $-15.8428$ | $-1.0815$
| Valence-Core |  | $0.0010$ | $-0.0024$ | $-0.0355$ |  | $-0.0017$ | $0.0045$ | $0.0664$
| Core-Valence |  | $0.0010$ | $-0.0024$ | $-0.0355$ |  | $-0.0017$ | $0.0045$ | $0.0664$
| Core-Core |  | $\sim 0.0$ | $\sim 0.0$ | $\sim 0.0$ |  | $\sim 0.0$ | $\sim 0.0$ | $\sim 0.0$
| Core |  | $\sim 0.0$ | $\sim 0.0$ | $\sim 0.0$ |  | $\sim 0.0$ | $\sim 0.0$ | $\sim 0.0$

It is possible to improve only the valence contributions to
$\alpha_{J}^{S}(\omega)$ and $\alpha_{J}^{A}(\omega)$ in the aforementioned
approach as only the E1 matrix elements involving the bound excited states can
be evaluated using the RCC method. However, correlation contributions
involving core excitations to the core and core-valence Goldstone diagrams
shown as Figs. 1(i) and (ii) have to be obtained from the first-principle
calculations. We have employed RPA to evaluate the core and core-valence
contributions to $\alpha_{J}^{S}(\omega)$ and $\alpha_{J}^{A}(\omega)$. In
both cases, we rewrite the expressions for both $\alpha_{J}^{S}(\omega)$ and
$\alpha_{J}^{A}(\omega)$ in a general form as

$\displaystyle\alpha_{J}^{\cal K}$ $\displaystyle=$
$\displaystyle\langle\Phi_{0}|D|\Phi_{0}^{(\infty,1)+}\rangle+\langle\Phi_{0}|D|\Phi_{0}^{(\infty,1)-}\rangle,$
(83)

where ${\cal K}$ stands either for $S$ (scalar) or for $A$ (axial-vector) and
$|\Phi_{0}^{(\infty,1)\pm}\rangle$ are the perturbation wave functions with
respect to the DHF wave function $|\phi_{0}\rangle$ for $\pm\omega$ values at
the energy denominator. These perturbative wave functions contain core-
polarization effects to all-orders and one-order of external dipole
interaction. It should be noted that for the scalar and axial-vector
components the corresponding angular factors are included but not shown
explicitly in the above expression.

Since experimental value for $\alpha_{J}^{S}(0)$ of the ground state of 133Cs
is known very precisely, comparison between our calculation with the
experimental result will help to validate our calculations for the dynamic
values of $\alpha_{J}^{S}(\omega)$ and $\alpha_{J}^{A}(\omega)$. Also, this
test would be useful for determining hyperfine-induced third-order
polarizabilities. In Figs. 2, 3 and 4, we show the Goldstone diagram
representations of all possible contributions to the DHF values of
$\alpha_{F}^{S/A/T(2,1)}$ for the top, center, and normalization contributions
respectively. These contributions are much smaller than the second-order
contributions to $\alpha_{F,M_{F}}$, but their accurate evaluations are more
challenging than the second-order contributions. For easy understanding of
various contributions to these quantities, we denote contributions from Fig. 2
(i) and (ii) together as core, (iii) as core-core, (iv) as core-valence, (v)
as valence-core, and (vi) as valence contributions. Analogous division has
been followed for diagrams shown in Fig. 3 as both figures 2 and 3 have
striking similarities. In Fig. 4, diagram (i) is denoted as core, diagram (ii)
as valence-core, and diagram (iii) as valence contributions as in the case of
the second-order E1 polarizabilities.

|  |
---|---|---
(a) | (b) | (c)
Figure 5: Demonstration of contributions from two different combinations of
intermediate states ($J^{\prime}$ and $J^{\prime\prime}$) to the (a) top, (b)
center and (c) normalization parts of the static $\alpha_{F}^{S(2,1)}$ value
of the $F=3$ level of 133Cs. States with subscript $-$ symbol in the figure
represent the lower angular momentum state of a fine-structure partner; i.e.
$P_{\\_}$ means $P_{1/2}$ and $D_{\\_}$ denotes $D_{3/2}$, while $P$ and $D$
stand for the $P_{3/2}$ and $D_{5/2}$ states respectively.

We adopt similar procedures of evaluating the second-order E1 polarizabilities
to estimate the valence contributions to $T^{\cal K}$, $C^{\cal K}$, and
$R^{\cal K}$. As can be seen in Fig. 2, estimation of the valence contribution
to $T^{\cal K}$ requires a large number of matrix elements involving the
$S_{1/2}$, $P_{1/2;3/2}$, and $D_{3/2}$ states. Unlike the second-order
polarizabilities, knowing correct signs for the E1 and $T_{J}^{(1)}$ matrix
elements are essential for the evaluation of $T^{\cal K}$. Evaluation of the
valence contribution to $C^{\cal K}$, requires E1 matrix elements for
transitions from the ground state to the $P_{1/2;3/2}$ states and
$T_{J}^{(1)}$ matrix elements for transitions between the $P_{1/2;3/2}$ states
as per the parity and angular momentum selection rules. Since the expressions
for $R^{\cal K}$ and second-order polarizability have similar forms, its
valence contribution evaluation requires the same E1 matrix elements as the
case of the second-order E1 polarizabilities along with the expectation value
of $T_{J}^{(1)}$ in the ground state.

It is important to consider the core, core-core, core-valence, and valence-
core contributions to $T^{\cal K}$ and $C^{\cal K}$ judiciously in order to
claim accuracy of the third-order E1 polarizability calculations. The core and
valence-core contributions to $R^{\cal K}$ are determined by adopting the same
approaches as mentioned earlier in the case of the second-order E1
polarizabilities. Unlike for $R^{\cal K}$, the core, core-core, core-valence,
and valence-core contributions to $T^{\cal K}$ and $C^{\cal K}$ have to be
estimated very carefully. As can be seen from Figs. 2 and 3, the core
contributions to these quantities require matrix elements involving the core-
core, core-virtual, and virtual-virtual orbitals. It is evident that
evaluations of the core-valence and valence-core contributions require similar
sets of matrix elements. However, different sets of core and virtual orbitals
are involved in the determination of the core and valence contributions to
$T^{\cal K}$ and $C^{\cal K}$ owing to different angular momentum selection
rules in both the expressions. Matrix elements between the bound states are
taken from the RCC theory or experiments as appropriate depending on their
accuracy. We also use here the experimental energies in the denominator
wherever possible, otherwise, the calculated energies are being used. The E1
matrix elements between the core orbitals are taken from the DHF method, while
between the core and virtual orbitals are taken from RPA as required.

|
---|---
(a) | (b)
Figure 6: Contributions from different combinations of intermediate states ($J^{\prime}$ and $J^{\prime\prime}$) to the (a) top and (b) center parts of the static $\alpha_{F}^{T(2,1)}$ value of the $F=3$ level of 133Cs. The notation is same as in the previous figure. Table 6: The ‘main’ contributions of $T(\omega)$, $C(\omega)$ and $R(\omega)$ to the $\alpha_{F}^{S(2,1)}$, $\alpha_{F}^{A(2,1)}$, $\alpha_{F}^{T(2,1)}$ values of the $F=3$ and $F=4$ hyperfine levels of the ground state of 133Cs at different wavelengths. All values are in a.u.. | $\lambda=\infty$ |  |  $\lambda=936$ nm |  |  $\lambda=1064$ nm
---|---|---|---|---|---
Contribution |  $\alpha_{F}^{S(2,1)}$ |  $\alpha_{F}^{A(2,1)}$ |  $\alpha_{F}^{T(2,1)}$ |  |  $\alpha_{F}^{S(2,1)}$ |  $\alpha_{F}^{A(2,1)}$ |  $\alpha_{F}^{T(2,1)}$ |  |  $\alpha_{F}^{S(2,1)}$ |  $\alpha_{F}^{A(2,1)}$ |  $\alpha_{F}^{T(2,1)}$
For $F=3$ level
$T(\omega)$ |  $-0.00121$ |  $0.0$ |  $0.00002$ |  |  $-0.00976$ |  $0.00145$ |  $0.00020$ |  |  $-0.00376$ |  $0.00012$ |  $0.00006$
$C(\omega)$ |  $0.00001$ |  $0.0$ |  $-0.00009$ |  |  $-0.01137$ |  $0.03754$ |  $-0.02486$ |  |  $-0.00031$ |  $0.00367$ |  $-0.00180$
$R(\omega)$ |  $-0.00376$ |  $0.0$ |  $0.0$ |  |  $-0.49459$ |  $0.18917$ |  $0.0$ |  |  $-0.05755$ |  $0.00794$ |  $0.0$
For $F=4$ level
$T(\omega)$ |  $0.00083$ |  $0.0$ |  $-0.00003$ |  |  $0.00670$ |  $0.00127$ |  $-0.00029$ |  |  $0.00258$ |  $0.00010$ |  $-0.00009$
$C(\omega)$ |  $-0.00001$ |  $0.0$ |  $0.00013$ |  |  $0.00780$ |  $0.03325$ |  $0.03703$ |  |  $0.00021$ |  $0.00325$ |  $0.00268$
$R(\omega)$ |  $0.00258$ |  $0.0$ |  $0.0$ |  |  $0.33926$ |  $0.16752$ |  $0.0$ |  |  $0.03948$ |  $0.00703$ |  $0.0$

##  IV Results and Discussion

In Tables 1 and 2, we present the $\alpha_{J}^{S}$, $\alpha_{J}^{A}$,
$\alpha_{F}^{S(2,1)}$, $\alpha_{F}^{A(2,1)}$ and $\alpha_{F}^{T(2,1)}$ values
of the $6S$ state of 133Cs at different wavelengths. We have used
$g_{I}=0.737885714$ with $I=7/2$ from Ref. [49] for carrying out these
evaluations. To understand the importance of the correlation effects and
sensitivity of the results due to use of the calculated and experimental
energies, we have given ab initio results from the DHF, RCCSD, and RCCSDT
methods in the tables. However, we give our final recommended values from the
semi-empirical approach after utilizing experimental energies and E1 matrix
elements as discussed in the previous section. These recommended results,
shown in bold font in the above tables, are compared with the available
experimental results and some of the previous calculations from the
literature. As can be seen from these tables, there are significant
differences between the DHF values and the RCCSD results. This suggests that
the electron correlations play significant roles in the accurate determination
of both the second-order and third-order E1 polarizabilities. These
differences are more prominent in the dynamic E1 polarizabilities. In fact,
there are sign differences between the DHF and RCCSD values from the atomic
polarizabilities indicating that correlation contributions are unusually large
in these quantities. By analysing the DHF and RCCSD results carefully, we
observe that large differences in these results are mostly due to the energy
denominators. This justifies the reason why the results are improved
significantly when experimental energies are used. Though differences among
the ab initio results and the semi-empirical values reduce when correlation
effects through triple excitations are included in the calculations, there are
still significant differences between the RCCSDT and semi-empirical values for
the dynamic polarizabilities. Since our objective is to offer precise values
of the E1 polarizabilities of the hyperfine levels of the ground state in
133Cs, the semi-empirical results are recommended for their future
applications. At this stage, we would like to clarify that only the valence
contributions are improved through the semi-empirical approach but the core,
core-valence, and valence-core contributions are taken from our calculations.
Thus, there is still scope to improve accuracy of the calculated results by
including higher-order correlation effects in the determination of the core,
core-core, core-valence, and valence-core contributions. Nonetheless,
uncertainties of our semi-empirical values quoted in both Tables 1 and 2
include typical orders of magnitudes from these neglected contributions.

Comparison of the static $\alpha_{J}^{S}$ and $\alpha_{F}^{T(2,1)}$ values
with their experimental results shows that our recommended values agree
perfectly with the measurements [53, 34]. Compared to the previous
calculations of the static $\alpha_{J}^{S}$ values reported in Refs. [25, 50,
51, 52], our value is very close to the experimental result. This is owing to
the fact that we have used many precisely estimated E1 matrix elements from
the latest measurements [54, 55] as discussed later. From this, we expect that
our other calculated values including the dynamic polarizabilities at
wavelengths $936$ nm and $1064$ nm are also equally accurate. We could not
find experimental results for $\alpha_{F}^{S(2,1)}$ and $\alpha_{F}^{A(2,1)}$
for either the $F=3$ level or the $F=4$ level to make direct comparison with
our estimated values. However, comparison with another calculation reported in
Ref. [33] show that the results for $\alpha_{F}^{S(2,1)}$ agree reasonably but
they differ significantly for $\alpha_{F}^{T(2,1)}$. In Ref. [33], the authors
have employed the combined TDHF and BO (TDHF$+$BO) method that accounts for
core-polarization effects to all-orders while pair-correlation contributions
have been estimated using the Brückner orbitals. The RCC method includes all
the RPA effects and pair-correlations to all-orders implicitly. We have come
across another semiempirical calculation wherein the authors employed the
Relativistic Configuration Interaction plus Core Polarization (RCICP) method
to compute the values of $\alpha_{F}^{S(2,1)}$ and $\alpha_{F}^{T(2,1)}$ [25].
Notably, there exist significant disparities between our calculated results
and theirs. We also found another semi-empirical result for
$\alpha_{F}^{T(2,1)}$ for the $F=4$ level [31], in which the calculation was
performed by using the statistical Thomas-Fermi potential approach and by
scaling some of the matrix elements with the experimental data. It has
overestimated the $\alpha_{F}^{T(2,1)}$ value compared to the experimental
result and also differs from our calculation.

After discussing the final results, we intend now to analyze individual
contributions to the final results to understand their roles for accurate
determination of both the second-order and third-order E1 polarizabilities.
Intermediate contributions to $\alpha_{J}^{S}(\omega)$ and
$\alpha_{J}^{A}(\omega)$ at different $\omega$ (rather $\lambda$) values are
given in Table 3. It lists the E1 matrix elements of many important
transitions that give dominant contributions to the valence part and are
referred as ‘main’. As mentioned before, many of these E1 matrix elements are
borrowed from the precise measurements of lifetime or E1 polarizability in
different atomic states that are reported in Refs. [54, 55]; others are taken
from the present RCCSDT method. The “tail” contributions to the valence part
from the high-lying virtual states are estimated by using the E1 matrix
elements from the DHF method and energies from the NIST database. The core and
core-valence contributions are estimated using RPA. It shows that precise
estimate of the second-order E1 polarizabilities depends mainly on the
accurate E1 matrix elements of the $6s~{}^{2}S_{1/2}\rightarrow
6p~{}^{2}P_{1/2;3/2}$ transitions and core contribution. However,
contributions from the E1 matrix elements of the $6s~{}^{2}S_{1/2}\rightarrow
7p~{}^{2}P_{1/2;3/2}$ transitions are also important to consider for improving
the precision of the results.

We discuss then the $\alpha_{F}^{S(2,1)}$, $\alpha_{F}^{A(2,1)}$ and
$\alpha_{F}^{T(2,1)}$ contributions to both the $F=3$ and $F=4$ hyperfine
levels at different wavelengths. As mentioned in the previous section, these
calculations require a large set of E1 and $T_{J}^{(1)}$ matrix elements. Some
of the dominantly contributing E1 matrix elements used in these calculations
are already given in Table 3. In Table 4, we list many $T_{J}^{(1)}$ matrix
elements that are important for the evaluation of $\alpha_{F}^{S(2,1)}$,
$\alpha_{F}^{A(2,1)}$ and $\alpha_{F}^{T(2,1)}$. Most of these results are
obtained using the RCCSDT method, except in a few cases for which we use the
precise values from the experiments [56, 57, 58, 59, 60, 61]. Some of the off-
diagonal matrix elements from this list are inferred from the experimental M1
hyperfine structure constants by using the relation

$\displaystyle\langle J_{f}||T_{J}^{(1)}||J_{i}\rangle\simeq\sqrt{\langle
J_{f}||T_{J}^{(1)}||J_{f}\rangle\langle J_{i}||T_{J}^{(1)}||J_{i}\rangle}.$
(84)

We have also used the experimental energies [44] wherever possible in order to
reduce uncertainties in the calculations.

Following the previous section discussion, these quantities are estimated by
dividing their contributions into $T^{\cal K}$, $C^{\cal K}$, and $R^{\cal
K}$. Further, each of these has core, core-core, core-valence, valence-core,
and valence contributions. Table 5 gives the individual contributions from the
core, core-core, core-valence, valence-core, and valence parts to the
$\alpha_{F}^{S(2,1)}$, $\alpha_{F}^{A(2,1)}$ and $\alpha_{F}^{T(2,1)}$ values
obtained by adding them from $T^{\cal K}$, $C^{\cal K}$ and $R^{\cal K}$
separately. It is evident from Table 5 that the valence contributions are the
dominant ones in the final values, whereas in $\alpha_{F}^{S(2,1)}$ and
$\alpha_{F}^{A(2,1)}$, contributions from the core, core-core, core-valence
and valence-core parts are negligibly small. One should also note that
contributions from the valence-core or core-valence correlations to the tensor
polarizabilities are non-negligible. Since an experimental result for the
static $\alpha_{F}^{T(2,1)}$ value of the $F=4$ level in 133Cs is available,
we intend to analyze it in terms of different correlation contributions. It is
evident from Table 5 that the valence contribution to this quantity from our
calculation is $-3.08\times 10^{-12}$ Hz/(V/m)2, whereas the central value of
the experimental result is $-3.34\times 10^{-12}$ Hz/(V/m)2 [34]. Thus, there
is about 8% difference between the two values after neglecting their
uncertainties. Reducing uncertainty due to systematic effects in the
measurement of $\alpha_{F}^{T(2,1)}$ would be extremely difficult, so it is
important to figure out roles of other physical contributions to the
theoretical result in order to help future experiments to carry out the
measurement more precisely. Our analysis shows that the core and core-core
contributions to the static $\alpha_{F}^{T(2,1)}$ value of the $F=4$ level are
negligibly small, while the valence-core and core-valence contributions are
quite significant. As can be seen from the table, the difference between the
theoretical and experimental value reduces drastically to 2% after taking into
account these contributions. Interestingly, these valence-core and core-
valence contributions to the dynamic $\alpha_{F}^{T(2,1)}$ values at
$\lambda=936$ nm and $\lambda=1064$ nm are found to be extremely small
compared to their valence contributions.

Unlike the second-order E1 polarizabilities, it is not possible to demonstrate
contributions from the intermediate states easily as their formulas possess
two summations (see Eqs. (55) and (63)). However, we adopted a different
approach to show importance of contributions from various intermediate states.
Figs. 5 and 6 present three-dimensional plots depicting contributions from two
different sets of intermediate states to the valence parts of $T^{\cal K}$,
$C^{\cal K}$, and $R^{\cal K}$ to the static $\alpha_{F}^{S(2,1)}$ and
$\alpha_{F}^{T(2,1)}$ values respectively. They are shown only for the $F=3$
level as a representative case. As can be seen from these figures, matrix
elements of a few selective transitions involving combinations of a few
selective intermediate states are contributing predominantly to the third-
order E1 polarizabilities. Gaining this knowledge is quite important in order
to improve precision of these quantities further. It is evident from Fig. 5
that the $6P_{1/2,3/2}$ and $7S_{1/2}$ intermediate states make the largest
contributions to the top, center and normalization parts of
$\alpha_{F}^{S(2,1)}$. However, significant contributions to the top and
center parts of $\alpha_{F}^{T(2,1)}$ come from $6P_{1/2,3/2}$ and $5D_{3/2}$
states, as seen in Fig. 6. Having clarified the roles of different
intermediate states in the determination of the third-order E1
polarizabilities, we present the main contributions to both the static and
dynamic $T^{\cal K}$, $C^{\cal K}$, and $R^{\cal K}$ values of
$\alpha_{F}^{S(2,1)}$, $\alpha_{F}^{A(2,1)}$ and $\alpha_{F}^{T(2,1)}$ by
taking sums of total contributions from all possible intermediate states in
Table 6. As can be seen from the table, the $R^{\cal K}$ component exhibits
the dominant contribution to $\alpha_{F}^{S(2,1)}$ followed by $T^{\cal K}$
and then the $C^{\cal K}$ component. For $\alpha_{F}^{A(2,1)}$ also $R^{\cal
K}$ contribution dominates, followed by the $C^{\cal K}$ part. In the case of
$\alpha_{F}^{T(2,1)}$, the leading contribution comes from the $C^{\cal K}$
part, while the $R^{\cal K}$ component is zero.

Table 7: Summary of the $k_{s}$ value from different theoretical and experimental works in units of $10^{-10}$ Hz/(V/m)2. Reference |  $k_{s}$ value
---|---
This work | $-2.274(10)$
Theory [19] | $-1.97(9)$
Theory [20] | $-2.06(1)$
Theory [21] | $-2.281(4)$
Theory [22] | $-2.28$
Theory [23] | $-2.26(2)$
Theory [24] | $-2.271(8)$
Theory [25] | $-2.324(5)$
Theory [33] | $-2.26(2)$
Experiment [26] | $-2.271(4)$
Experiment [27] | $-2.25(5)$
Experiment [28] |  $-2.20(26)$*
Experiment [29] |  $-1.89(12)$*
Experiment [30] | $-2.05(4)$
* $k_{s}$ calculated from BBR shift measurement. |

In Table 7, we present a comparison between our calculated Stark shift
coefficient,
$k_{s}=-\frac{1}{2}\big{(}\alpha_{F=4}^{S(2,1)}-\alpha_{F=3}^{S(2,1)}\big{)}$,
and the previously reported values. As can be seen from the table, our value
$-2.274(10)\times 10^{-10}$ Hz/(V/m)2 closely aligns with the most precise
measurement to date, which is reported as $-2.271(4)\times 10^{-10}$ Hz/(V/m)2
in Ref. [26]. It also aligns with other experimental values in Refs. [27] and
[28]. In contrast, it differs substantially from other measurements reported
later in Refs. [29, 30]. We are unable to provide insights regarding the
discrepancies among experimental results. Nevertheless, we have thoroughly
examined and discussed the differences observed among the theoretical results.
We find that our result as precise as the calculated value reported in Ref.
[24]; their and our result agree better with the experiment [26] compared to
other theoretical works [19, 20, 21, 22, 23, 25, 33]. This may be due to our
semi-empirical treatment of various contributions to the estimations of the
$\alpha_{F=3}^{S(2,1)}$ and $\alpha_{F=4}^{S(2,1)}$ values. Also, our DHF
value $-2.792\times 10^{-10}$ Hz/(V/m)2 of $k_{s}$ agrees with the DHF value
$-2.799\times 10^{-10}$ Hz/(V/m)2 of Ref. [24]. Again, authors of Ref. [24]
have found the contributions to $k_{s}$ arising from the continuum (tail) to
be significant. In this work, we also independently verify this finding and
affirm that without the tail contribution the $k_{s}$ value comes out to be
$-2.085\times 10^{-10}$ Hz/(V/m)2. One can infer these tail contributions from
our calculations to the hyperfine interaction induced E1 polarizabilities
explicitly by analyzing various contributions listed in Tables 5 and 6. It can
be seen from these tables that the tail contribution to $k_{s}$ comes out to
be 8% to the total contribution and the largest uncertainty in our final
$k_{s}$ value arises mainly from this part.

##  V Summary

We have conducted comprehensive analyses of the second-order and magnetic
dipole hyperfine interaction induced third-order electric dipole
polarizabilities of the hyperfine levels of the ground state of the 133Cs
isotope. Results are presented for the DC electric field and for the AC
electric field with two different wavelengths. One of them corresponds to the
magic wavelength of the cooling line of the 133Cs atom, but power of laser
available at this wavelength is usually very low. There exist high-power
lasers for the other chosen wavelength; such lasers are often used in high-
precision laboratory measurements. First, we present the second-order electric
dipole polarizabilities and compare them with the precisely reported
experimental value and other theoretical results. After validating
calculations through these results, we proceeded with the determination of the
magnetic dipole hyperfine interaction induced third-order electric dipole
polarizabilities. In order to understand these results thoroughly, we gave a
breakdown of the results in terms of contributions from intermediate states
involving both the core and valence orbitals. Our static values for both the
second-order and third-order electric dipole polarizability values match with
the available experimental results quite nicely and explain the roles of
various contributions to accurate evaluation of these quantities. The reported
static and dynamic electric dipole polarizability results for both hyperfine
levels of the ground state in 133Cs can be immensely useful to the
experimentalists for estimating the Stark effects precisely to carry out high-
precision laboratory measurements.

## Acknowledgment

The computations reported in the present work were carried out using the
ParamVikram-1000 HPC cluster of the Physical Research Laboratory (PRL),
Ahmedabad, Gujarat, India

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