# Vector polarizability of atomic state induced by a linearly polarized vortex
beam: External control of magic, tune-out wavelengths, and heteronuclear spin
oscillations

Anal Bhowmik<EMAIL_ADDRESS>Haifa Research Center for
Theoretical Physics and Astrophysics, University of Haifa, Haifa 3498838,
Israel  Department of Mathematics, University of Haifa, Haifa 3498838, Israel
Narendra Nath Dutta  Department of Chemical Sciences, Indian Institute of
Science Education and Research Mohali, Punjab-140306, India.  Sonjoy Majumder
<EMAIL_ADDRESS>Department of Physics, Indian Institute of
Technology Kharagpur, Kharagpur-721302, India.

###### Abstract

Experiments with vortex beams have shown a surge of interest in controlling
cold atoms. Most of the controlling protocols are dominated by circularly
polarized light due to its ability to induce vector polarization at atoms,
which is impossible for paraxial linearly polarized light. Here we develop a
theory for frequency dependent polarizability of an atomic state interacting
with a focused linearly polarized vortex beam. The naturally induced spin-
orbit coupling in this type of linearly polarized beam produces vector
component of the valence polarizability to an atomic state, obeying the total
angular momentum conservation of the beam. The theory is employed on 87Sr+ ion
to accurately calculate the magic wavelengths for the clock transitions and
tune-out wavelengths for the clock states using relativistic coupled-cluster
method. The induced vector component in the dynamic polarizability due to the
linearly polarized focused vortex beam promotes fictitious magnetic field to
the atomic state. We demonstrate that this fictitious magnetic field,
depending on the focusing angle and OAM of the beam, improves the flexibility
of the coherent heteronuclear spin oscillations in a spin-1 mixture of 87Rb
and 23Na atoms.

##  I INTRODUCTION

In recent years, laser trapping and cooling of neutral atoms have attracted
significant attention to the experimentalists and achieved as an established
technique in high precision spectroscopic measurements Champenois2004 ;
Chou2010 ; Ludlow2015 . Along with neutral atoms, optical dipole trappings of
ions have recently surged interest in ultra-cold physics Laupretre2019 ;
Schmidt2018 ; Lambrecht2017 ; Schaetz2017 . However, mechanism of the trapping
by optical means inevitably produces the Stark shifts in the atomic energy
levels and influences the fidelity of the precision measurements. Magic
wavelengths are the unique wavelengths of the external laser beam for which
the differential ac-Stark shift of an atomic transition effectively vanishes.
Therefore, the impediment in precise spectroscopic measurements can be
eliminated almost entirely if the atoms are confined at the predetermined
magic wavelengths of the laser beam. The light at magic wavelengths have
significant applications in atom optics, such as atomic interferometers
Biedermann2015 , atomic clocks Margolis2009 ; Rosenbusch2009 ; Nicholson2015 ,
and atomic magnetometers Dong2015 , etc.

Determinations of magic wavelengths for an atomic transition depend mainly on
how accurately the frequency-dependent valence polarizability (POL) values of
the atomic Zeeman sub-levels are calculated. The valence POL consists of three
components, scalar POL ($\alpha_{V}^{0}$), vector POL ($\alpha_{V}^{1}$), and
tensor POL ($\alpha_{V}^{2}$) Flambaum2008 ; Arora2007 ; Lacroute2012 ;
Arora2012 . In general, POL is defined in terms of an off-resonant electric
dipole interaction between the atom and its trapping light. The vector POL,
which yields the energy shift among the Zeeman sublevels, arises usually due
to a circularly polarized light Flambaum2008 ; Bhowmik2018 ; Das2020 . It is
one of the non-magnetic field sources to remove Zeeman degeneracy and becomes
very significant in the evaluation of magic wavelengths for cases such as
”knob” to adjust an optical trap Kim2013 .

In this work, we theoretically develop and demonstrate that a linearly
polarized focused vortex (LP-FV) beam can naturally produce vector part of the
dynamic POL when it interacts with cold atoms or ions. The vector part is
certainly not a component of dynamic POL for a linearly polarized Gaussian or
paraxial vortex beam. The distinctive feature of the vortex beam is that in
addition to the spin angular momentum associated with the polarization, it has
orbital angular momentum (OAM) which arises due to the helical phase front of
the beam Allen1992 ; Babiker2002 ; Lembessis2011 ; Afanasev2013 ; Mondal2014 ;
Bhowmik2016 ; Schmiegelow2016 ; Bhowmik_JPC_2018 ; Romero2012 ; SDas2020 ;
Quinteiro2020 . We find that the origin of the vector POL comes from the
prolific artifacts of the spin-orbit coupling of LP-FV beam satisfying the
conservation of the total angular momentum of the beam during the interaction-
process. Also, this vector POL creates a fictitious magnetic field Kien2013
when it interacts with an atomic system. This fictitious magnetic field
interrogates, manipulates, and traps an atom or ion independently Albrecht2016
or in conjunction with the real magnetic field Schneeweiss2014 . Further, this
fabricated field provides a unique opportunity for exploring spin-dynamics in
the form of coherent spin oscillations Stenger1998 ; Chang2004 ; Chang2005 ;
Klempt2009  in the ultra-cold spinor quantum gas. This has been the subject of
intensive theoretical and experimental research due to its high
controllability Kurn2013 .

The primary objective of the present work is to visualize the effect of vector
POL, originated due to the spin-orbit coupling of the LP-FV beam, in the clock
transitions and clock states. We apply our theory to calculate the dynamic POL
values of the fine-structure states, $5S_{\frac{1}{2}}$, $4D_{\frac{3}{2}}$,
and $4D_{\frac{5}{2}}$ of 87Sr+. Singly ionized strontium is an excellent
candidate for optical frequency standard, quantum information, and quantum
storage Akerman2015 ; Shapira2018 . As the spin-orbit coupling of LP-FV beam
varies with the choice of OAM and focusing angle of the lens, we
quantitatively show the effects of these parameters on the magic wavelengths
for $5S_{\frac{1}{2}}\rightarrow 4D_{\frac{3}{2},\frac{5}{2}}$ transitions and
tune-out wavelengths (where the dynamic POL value becomes zero LeBlanc2007 ;
Wang2016 ). These quantifications will be important for the experimentalists
to choose the parameters of the vortex beams used for trapping Kim2013 ;
Albrecht2016 . In order to manifest the impact of the vector POL, we compare
the calculated magic and tune-out wavelengths induced from the LP-FV beam with
the corresponding wavelengths induced from a linearly polarized Gaussian beam.
Moreover, as an important application, we investigate how the fictitious
magnetic field, originated from the vector POl, can control the coherent
heteronuclear spin oscillations in a spin-1 mixture of 87Rb and 23Na atoms
with influences from both linear and quadratic Zeeman shifts.

##  II THEORY

According to the time-independent second-order perturbation theory Mitroy2010
, the ac-Stark shift of an atomic state in an external oscillating electric
field $\mathcal{E}(\omega)$ is expressed by $\Delta
F(\omega)=-\frac{1}{2}\alpha^{T}(\omega)\mathcal{E}^{2}$, where
$\alpha^{T}(\omega)$ is the total dynamic POL of the atomic state at frequency
$\omega$ and $\mathcal{E}$ is the magnitude of the external electric field.
For a single-valence atom with a valence electron in the $v$th orbital, the
total dynamic POL can be written as
$\alpha^{T}(\omega)=\alpha^{C}(\omega)+\alpha^{VC}(\omega)+\alpha^{V}(\omega)$.
Here $\alpha^{C}(\omega)$ is the frequency-dependent core POL of the ionic
core obtained by removing the valence electron. $\alpha^{VC}(\omega)$ gives a
correction Dutta2015  to the core POL due to the presence of the valence
electron, and it is considered as $\omega$-independent in the present work.
$\alpha^{V}(\omega)$ is the valence POL of the single-valence state.
$\alpha^{C}$ and $\alpha^{VC}$ provide small contributions to $\alpha^{T}$
compared to $\alpha^{V}$, and they are computed approximately using lower-
order many-body perturbation theory Mitroy2010 ; Ghosh1993 ; Dutta2020 .

Now, we evaluate $\alpha^{V}(\omega)$ in the presence of an external LP-FV
beam. As Laguerre-Gaussian (LG) beam is a well-established example of a vortex
beam, initially, we assume a paraxial form of linearly polarized coherent LG
beam without any off-axis node and propagates along the $z$-axis. The field is
expressed as Mondal2014
$\bm{{{\mathcal{E}_{i}}}}(\rho,\phi,z,t)=\mathcal{E}_{i}(t)\left({{\sqrt{2}\rho}/{w_{0}}}\right)^{\lvert
l\rvert}e^{i(l\phi+k_{0}z)}\bm{\hat{\textbf{x}}}$. Here $k_{0}$ is the wave
number of the free space, $w_{0}$ is the waist-size and $l$ is OAM of the
beam. We consider that this paraxial LG beam is focused by passing through an
objective (lens) with a high numerical aperture (NA) Bhowmik2016 . Then this
focused LG beam interacts with a cold atom or ion whose de Broglie wavelength
is large enough to experience the intensity variation of this beam. In order
to take a full advantage of the high NA of the lens, we assume that $w_{0}$
overfills the entrance aperture radius. According to the Kirchhoff’s
approximation in diffraction theory Richards1959 ; Boivin1965 , the consequent
components of the spin-orbit coupled LP-FV beam can be expressed as

$\displaystyle\mathcal{E}$
$\displaystyle(\rho,\phi,z,t)=\left[\begin{array}[]{c}\mathcal{E}_{x}\\\
\mathcal{E}_{y}\\\
\mathcal{E}_{z}\end{array}\right]=(-i)^{l+1}\mathcal{E}_{0}e^{i(l\phi-\omega
t)}$ (4) $\displaystyle\times$
$\displaystyle\left[\begin{array}[]{c}u_{l}(\rho,z)+u_{l+2}(\rho,z)e^{2i\phi}+u_{l-2}(\rho,z)e^{-2i\phi}\\\
-i(u_{l+2}(\rho,z)e^{2i\phi}-u_{l-2}(\rho,z)e^{-2i\phi})\\\
-i(u_{l+1}(\rho,z)e^{i\phi}-u_{l-1}(\rho,z)e^{-i\phi})\end{array}\right].$ (8)

Here $\mathcal{E}_{x}$, $\mathcal{E}_{y}$, and $\mathcal{E}_{z}$ are the $x$-,
$y$-, and $z$\- components of the electric field, respectively. The amplitude
of the focused electric field is $\mathcal{E}_{0}=\dfrac{\pi
f}{\lambda}T\mathcal{E}_{i}$, where $\mathcal{E}_{i}$ is the amplitude of the
incident electric field, $T$ is the transmission coefficient of the objective,
and $f$ is its focal length related to $\rho$ by $\rho=f\sin\theta$ (Abbe sine
condition). The coefficients $u_{l+m}$, where $m$ takes the values 0, $\pm 1$,
and $\pm 2$ in the above expression, depend on the focusing angle of NA
($\vartheta_{m}$) following the relation Zhao2007
$u_{l+m}(\rho,z)=\int_{0}^{\vartheta_{m}}d\vartheta\left({\dfrac{\sqrt{2}\rho}{w_{0}}}\right)^{\lvert
l\rvert}\sin\vartheta\sqrt{\cos\vartheta}g_{\lvert
m\rvert}(\vartheta)J_{l+m}(k\rho\sin\vartheta)e^{ikz\cos\vartheta}.$ Here
$J_{l+m}(k\rho\sin\vartheta)$ represents cylindrical Bessel function and
$k=\mu k_{0}$ with $\mu$ as the refractive index of the lens medium.
$g_{|m|}(\vartheta)$ are the angular functions with
$g_{0}(\vartheta)=1+\cos\vartheta$, $g_{1}(\vartheta)=\sin\vartheta$, and
$g_{2}(\vartheta)=1-\cos\vartheta$.

Now, let us discuss about Eq. (4) in detail. Linearly polarized light can be
considered as the superposition of left ($\beta=+1$) and right ($\beta=-1$)
circularly polarized beams. Because of the focusing and the diffraction from
the edges of the aperture, each circularly polarized light ($\beta=\pm 1$) can
be decomposed into three sets of local polarizations ($\pm 1$, $\mp 1$, and
polarization along z-axis) Bhowmik2016 . Among these three sets of local
polarizations, the first set has equal amplitude ($u_{l}$) for $\pm 1$ local
polarizations. Therefore, after passing through the focusing lens, the
superposition of these local polarizations results linearly polarized beam
with the OAM similar to the OAM of the incident beam. However, in the case of
second set, different field amplitudes, $u_{l+2}$ and $u_{l-2}$, are generated
with two different local polarizations ($-1$ and $+1$, respectively) and
topological charges ($l+2$ and $l-2$, respectively, to conserve total angular
momentum) after passing through the lens. Therefore, the field gains two
opposite circular polarizations with different amplitudes and creates the
vector part of valence POL in interaction with an atom or ion. The third set
yields $u_{l+1}$ and $u_{l-1}$ fields with topological charges, $l+1$ and
$l-1$, respectively. However, both these fields are polarized along the $z$\-
direction, which is another interesting manifestation of focusing beam.
Nevertheless, using LP-FV beam presented in Eq. (4), $\alpha^{V}(\omega)$ of
the atomic state can be presented by

$\alpha^{V}(\omega)=C_{0}\alpha_{V}^{0}(\omega)+C_{1}\alpha_{V}^{1}(\omega)+C_{2}\alpha_{V}^{2}(\omega).$
(9)

Here the coefficients $C_{i}$s are expressed in the following forms:

$\displaystyle C_{0}$ $\displaystyle=$
$\displaystyle\\{u_{l}\\}^{2}+\\{u_{l+1}\\}^{2}+\\{u_{l-1}\\}^{2}+2[\\{u_{l+2}\\}^{2}+\\{u_{l-2}\\}^{2}],$
$\displaystyle C_{1}$ $\displaystyle=$
$\displaystyle\left[2\\{u_{l-2}\\}^{2}-2\\{u_{l+2}\\}^{2}\right]\times\left(\frac{M_{J_{V}}}{2J_{V}}\right),\text{and}$
(10) $\displaystyle C_{2}$ $\displaystyle=$
$\displaystyle\left[\\{u_{l}\\}^{2}-\\{u_{l+1}\\}^{2}-\\{u_{l-1}\\}^{2}+2\\{u_{l+2}\\}^{2}+2\\{u_{l-2}\\}^{2}\right]$
$\displaystyle\times$
$\displaystyle\left({(3M_{J_{V}}^{2}-J_{V}(J_{V}+1))}/{(2J_{V}(2J_{V}-1))}\right),$

where $J_{V}$ and $M_{J_{V}}$ are the total angular momentum and its magnetic
component for the single-valence atomic state $|\Phi_{V}\rangle$. The
mathematical expressions of the scalar, vector, tensor, and consequently the
valence POL at the fine- and hyperfine-structure levels are presented in the
Appendix. The calculation of $\alpha^{V}(\omega)$ Mitroy2010 ; Bhowmik2018  of
an atomic state directly depends on different combinations of the integrals
$u_{l+m}$. These integrals can be altered with the various choices of the
topological charges of the incident LG beam and the NA of the objective. As a
consequence, the total POL values of the atomic states and the magic
wavelengths for the transitions among them can be tuned externally using
different parameters of the beam.

##  III RESULTS AND DISCUSSIONS

As the electron-correlation most significantly affects $\alpha^{V}(\omega)$
part of the dynamic POL due to the loosely bound valence electron, the
accurate estimations of the scalar, vector, and tensor components of
$\alpha^{V}(\omega)$ for the different states require correlation exhaustive
many-body calculations Lindgren1985 ; Bishop1987 ; Lindgren1986 ;
Lindgren_1985 ; Bartlett2007 ; Dutta2016 ; Bhowmik2017 ; Bhowmik2017b ;
Das2018  with a sophisticated numerical approach (see Appendix for the details
of computations). To have a quantitative analysis of the effect of OAM of LP-
FV beam on the total vector POL ($C_{1}\alpha_{V}^{1}(\omega)$) of an atomic
state, we show variations of $C_{1}\alpha_{V}^{1}(\omega)$ with frequency for
$5S_{\frac{1}{2}}(+1/2)$ and $4D_{\frac{3}{2},\frac{5}{2}}(+1/2)$ states of
87Sr+ in Fig. 1 ($M_{J}=-\frac{1}{2}$ provides opposite sign to
$C_{1}\alpha_{V}^{1}(\omega)$). The fine-structure states $5S_{\frac{1}{2}}$,
$4D_{\frac{3}{2}}$, and $4D_{\frac{5}{2}}$ are indicated by $5S1$, $4D3$, and
$4D5$, respectively in this figure and the next two figures of this paper.
Here LP-FV beam has either OAM=+1 or +2 with focusing angle of 60∘. Although
the effect of variation of focusing angle to $C_{1}\alpha_{V}^{1}(\omega)$ is
marginal, it is significant to the total scalar and tensor POL values, and
consequently to the total POL values. The peak values of
$C_{1}\alpha_{V}^{1}(\omega)$ occur at resonance frequencies: 0.11 a.u. for
$5S_{\frac{1}{2}}$ state; around 0.045 a.u and 0.21 a.u. for
$4D_{\frac{3}{2},\frac{5}{2}}$ states, respectively. However, many small scale
structures in $C_{1}\alpha_{V}^{1}(\omega)$ profiles of
$4D_{\frac{3}{2},\frac{5}{2}}$ states appear due to multiple number of
resonance transitions at around 0.25 a.u. to 0.30 a.u.

Figure 1:  Frequency dependent total vector POL ($C_{1}\alpha_{V}^{1}$) of
$5S_{\frac{1}{2}}(+1/2)$ and $4D_{\frac{3}{2},\frac{5}{2}}(+1/2)$ states for
LP-FV beam with OAM=+1 and +2 at focusing angle 60∘.

It can be seen from the graphs that OAM=+1 always produces higher peak value
of $C_{1}\alpha_{V}^{1}(\omega)$ compared to OAM=+2. This is because of the
stronger spin-orbit coupling for OAM=+1 compared to OAM=+2. Also, the
magnitude of the peak value of $C_{1}\alpha_{V}^{1}(\omega)$ is higher for
larger $M_{J}$ for a fixed value of $J$, which is obvious from the expression
of $C_{1}$.

Figure 2:  Frequency dependent $\alpha^{T}$ of $5S_{\frac{1}{2}}$ and
$4D_{\frac{3}{2},\frac{5}{2}}$ states. LP-FV beam is focused at 60∘ with
OAM=+1. The values inside the parentheses indicate the magnetic components.
The corresponding wavelengths at the intersections of the polarizability
curves of $5S_{\frac{1}{2}}$ and $4D_{\frac{3}{2},\frac{5}{2}}$ states are the
magic wavelengths for the clock transitions
$5S_{\frac{1}{2}}-4D_{\frac{3}{2}}$ and $5S_{\frac{1}{2}}-4D_{\frac{5}{2}}$.
For both these clock transitions, five sets (Set-1 to Set-5) of magic
wavelengths are found.

Fig. 2 displays profiles of $\alpha^{T}$ for $5S_{\frac{1}{2}}$ and
$4D_{\frac{3}{2},\frac{5}{2}}$ states of 87Sr+ at 60∘ focusing angle for LP-FV
beam with OAM=+1. However, it should be mentioned that focusing angle does
affect $\alpha^{T}$ values at various frequencies of the beam. Fig. 2 shows a
number of intersection points between $\alpha^{T}$ profiles of the multiplets
of $5S_{\frac{1}{2}}$ and $4D_{\frac{3}{2},\frac{5}{2}}$ states. These
intersection points indicate the magic wavelengths at which the differential
ac-Stark shift of the associated clock transition states vanishes. We observe
five sets (Set-1 to Set-5) of magic wavelengths for
$5S_{\frac{1}{2}}\rightarrow 4D_{\frac{3}{2},\frac{5}{2}}$ clock transitions
within the frequency span (from the near-infrared to the ultra-violet regions)
as shown in the figures (see Appendix). This is true for all the magnetic
sublevels involed in the transitions. However, the infrared magic wavelengths
(fall under Set-1 and Set-2) are the most important to trap 87Sr+ ion due to
relatively large $\alpha^{T}$ values and support the red-detuned trapping
scheme useful for frequency standard experiment. We find that the magic
wavelengths for the clock transitions at Set-1 are maximally affected by the
spin-orbit coupling of the beam. For $5S_{\frac{1}{2}}\rightarrow
4D_{\frac{5}{2}}(+3/2)$ transition, the magic wavelength at Set-1 is 1793.83
nm for either linearly polarized paraxial LG beam or Gaussian beam. The
results are same as OAM of the paraxial light does not affect the electronic
motion of an cold atom or ion (which is below its recoil limit) at the dipole
transition level Bhowmik2018 ; Mondal2014 . However, we observe that for
OAM=+1 and +2 of LP-FV beam the magic wavelengths of
$5S_{\frac{1}{2}}\rightarrow 4D_{\frac{5}{2}}(+3/2)$ transition at Set-1 can
be varied upto -20% and -15%, respectively, compared to the corresponding
magic wavelength of Gaussian beam.

Figure 3:  Variation of $\alpha^{T}$ (in a.u.) with wavelength (in nm) for all
the magnetic components of the state $4D_{\frac{3}{2}}$ for LP-FV beam focused
at 60∘ (with (a) OAM=+1 and (b) OAM=+2) and paraxial Gaussian beam are
presented. Wavelengths at which the curves intersects zero-polarizability axis
are the tune-out wavelengths.

Figure 4: (a) Variation of $\alpha^{T}$ (in a.u.) with wavelength (in nm) for
the ground state of the 87Rb at the hyperfine level $F_{V}=1$ for the LP-FV
beam focused at 60∘. (b) Magnetic energy diagram for the two heteronuclear
spin-oscillation processes without light shift:
$|0,0\rangle\leftrightarrow|-1,1\rangle$ (red dashed) and
$|0,0\rangle\leftrightarrow|1,-1\rangle$ (red solid). Blue solid and green
solid curves are magnetic energy variations with OAM=+1 and +2, respectively,
for $|0,0\rangle\leftrightarrow|1,-1\rangle$. The intensity of the LP-FV beam
is assumed to be 10 W/m2.

Fig. 3 illustrates wavelength dependence on $\alpha^{T}$ for the clock
transition state $4D_{\frac{3}{2}}$ of 87Sr+ due to the external field of LP-
FV beam with OAM=+1 and +2, focused at angle of 60∘, and linearly polarized
paraxial Gaussian beam. As the beam is linearly polarized, the vector
component does not arise in $\alpha^{T}$ for the Gaussian beam. Therefore, the
tune-out wavelengths for $4D_{\frac{3}{2}}(M_{J}=\pm 1/2)$ and
$4D_{\frac{3}{2}}(M_{J}=\pm 3/2)$ states are degenerated with respect to the
sign of $M_{J}$ and their values are 507.83 nm and 744.12 nm, respectively.
These degeneracies are lifted for LP-FV beam. This leads to splitting of the
tune-out wavelengths for the oppositely signed $M_{J}$-levels with respect to
the corresponding tune-out wavelengths for the linearly polarized Gaussian
beam. The figure shows that non-zero OAM of the LP-FV beam causes red-shifting
and blue-shifting of the tune-out wavelengths with respect to the tune-out
wavelengths of the Gaussian beam for $M_{J}=|\frac{1}{2}|$ and
$M_{J}=|\frac{3}{2}|$, respectively. The comparison of panels (a) and (b) of
Fig. 3 reveals that the separation between the tune-out wavelengths
corresponding to $+M_{J}$ and $-M_{J}$ of a particular value of $M_{J}$
decreases with the increasing OAM of the beam. This also indicates the
decreasing strength of the spin-orbit coupling with the increasing OAM of the
LP-FV beam

We also gauge the effect of the vector POL arising from LP-FV beam by
considering a heteronuclear spin-1 mixture of 87Rb and 23Na atoms with spin
states $|M_{F_{V1}}\rangle$ and $|M_{F_{V2}}\rangle$, respectively, of
$F_{V1}=F_{V2}=1$ hyperfine levels of their respective ground states
$5S_{\frac{1}{2}}$ and $3S_{\frac{1}{2}}$. Here, we show how their coherent
spin oscillations can be controlled by the fictitious magnetic field generated
from the vector POL due to LP-FV beam. We calculate the total magnetic energy
$E^{|M_{F_{V1}},M_{F_{V2}}\rangle}(B)$ (from the Breit-Rabi formula Vanier1988
) associated with $|M_{F_{V1}},M_{F_{V2}}\rangle$ state, where $B$ is the
externally applied magnetic field. In this example, we start with
$|0,0\rangle$ mixture of 87Rb and 23Na atoms. Therefore, we have only two
types of spin-changing transition process at $F_{V}=1$ level conserving the
total magnetization: $|0,0\rangle\leftrightarrow|-1,1\rangle$ and
$|0,0\rangle\leftrightarrow|1,-1\rangle$. In the presence of a laser beam, the
magnitudes of the frequency-dependent Stark-shifts for both the atoms largely
depend on the positions of the $D$-line transitions, which are close to 590 nm
and 790 nm for the 23Na and 87Rb atoms, respectively. Therefore, if a light
beam of wavelength nearly 790 nm is used, it can make the Stark-shift
essentially species-selective for 87Rb, but mostly transparent for 23Na. The
dynamic polarizability for the state $|5S_{1/2},F_{V}=1,M_{F_{V}}=0,\pm
1\rangle$ of 87Rb is depicted in Fig 4(a) for 786-794 nm. Here a LP-FV beam
with intensity $10$ W/cm2 and both OAM=+1, +2 are employed to selectively
dresses the energy levels of 87Rb near 790.14nm wavelength, which is the tune-
out wavelength for $M_{F_{V}}=0$ state. The figure shows that the Stark shift
for $M_{F_{V}}=\pm 1$ is larger for OAM=+1 compared to OAM=+2 at the
wavelength 790.14 nm.

Without the LP-FV beam, the total Zeeman energy difference of the process
$|0,0\rangle\leftrightarrow|-1,1\rangle$ is indicated by $\Delta
E^{-}(B)=E^{|0,0\rangle}-E^{|-1,1\rangle}$ and of the process
$|0,0\rangle\leftrightarrow|1,-1\rangle$ is indicated by $\Delta
E^{+}(B)=E^{|0,0\rangle}-E^{|1,-1\rangle}$. These Zeeman energy differences of
the heteronuclear spin-oscillations include contributions of both the linear
and quadratic Zeeman shifts Zhang2005 . Since the heteronuclear spin
oscillation can only occur near $\Delta E^{\pm}(B)=0$ Li2015 , Fig. 4(b)
suggests that only the oscillation $|0,0\rangle\leftrightarrow|1,-1\rangle$ is
possible at the magnetic field 0.99 G. The other oscillation,
$|0,0\rangle\leftrightarrow|-1,1\rangle$, is strongly suppressed at this
magnetic field. Now, taking into account the light shift by the LP-FV beam on
87Rb atom at the wavelength 790.14 nm, the total Zeeman energy difference of
the allowed process is modified to $\Delta
E^{+}(B)=(E^{|0,0\rangle}-E^{|1,-1\rangle})+[\delta E_{l}(M_{F_{V}}=0)-\delta
E_{l}(M_{F_{V}}=1)]$, where $\delta E_{l}$ is the light shift. In Fig 4(b), we
find that $\Delta E^{+}(B)$ becomes zero at the magnetic fields 1.40 G and
1.26 G for OAM=+1 and +2, respectively, of LP-FV beam. Therefore, the external
magnetic field that is required for $|0,0\rangle\leftrightarrow|1,-1\rangle$
transition with $\Delta E^{+}(B)=0$ to take place is changed significantly due
to the presence of the LP-FV beam, which accounts the fictitious magnetic
field generated by the vector polarizability. This is purely a manifestation
of the vector light shift arises from the spin-orbit coupling of the beam.
Also, the spin oscillations process of the heteronuclear spin-1 mixture of
87Rb and 23Na can be controlled externally by changing focusing angle and OAM
of LP-FV beam. This process will be very useful for recent experiment using
spherical-quadrupole magnetic trap Fang2019 .

##  IV CONCLUSION

In summary, we have developed a theoretical formalism to calculate the dynamic
POL of an atomic state in the external field of a LP-FV beam. We have
demonstrated how the OAM and focusing angle of the beam induce the spin-orbit
coupling in the beam, which in turn produces vector polarizability in the
atomic state even for linearly polarized light. The fine tuning of magic
wavelengths for the clock transitions $5S_{\frac{1}{2}}\rightarrow
4D_{\frac{3}{2},\frac{5}{2}}$ of 87Sr+ ion are demonstrated by controlling OAM
and focusing parameters of LP-FV beam (details are in the Appendix). The
vector POL values presented in this paper can be verified experimentally by
stimulated Raman spectroscopy Hu2018  or by measuring the tune-out wavelengths
for the magnetic sublevels of same multiplets Schmidt2016 ; Trubko2017 .
Moreover, we develop a controlling mechanism of heteronuclear spin
oscillations of spin-1 mixture of 87Rb and 23Na atoms with the fictitious
magnetic field generated by LP-FV beam. This mechanism can be used to tune the
heteronuclear spin dynamics Li2020  and generate entanglement between
distinguishable atoms Shi2006 . We believe that the present theoretical
development will give an additional flexibility in the spin-changing
interaction process of other multi-species spinor condensates.

## ACKNOWLEDGMENTS

A.B. acknowledges Arghya Das for helpful discussions.

## APPENDIX

In this Appendix, we augment the main text with further details. In section A,
we present a brief mathematical description of the scalar, vector, and tensor
polarizabilities of a single-valence atomic state at the fine-structure and
hyperfine levels. In section B, we show the numerical procedure to calculate
the dynamic polarizabilities of 87Sr+ and 87Rb. In this Section, we also
include the details of the Zeeman energy difference that leads the spin
oscillation processes of the spin-1 mixture 87Rb and 23Na atoms. In section C,
we tabulate a list of magic wavelengths and the corresponding vector and total
polarizabilities for the clock transitions $5S_{\frac{1}{2}}\rightarrow
4D_{\frac{3}{2},\frac{5}{2}}$ of 87Sr+ ion.

###  IV.1 Scalar, vector, and tensor polarizabilities at the fine-structure
and hyperfine levels

$\alpha_{V}^{0}(\omega)$, $\alpha_{V}^{1}(\omega)$, and
$\alpha_{V}^{2}(\omega)$ are the scalar, vector, and tensor components,
respectively, of the dynamic valence polarizability $\alpha^{V}(\omega)$ at
the fine-structure level of single-valence state $\Psi_{V}$. These components
are expressed using the sum-over-states approach Flambaum2008 ; Arora2007 ;
Lacroute2012 ; Arora2012 ; Mitroy2010 ; Dutta2015  as follows

$\alpha_{V}^{0}(\omega)=\frac{2}{3(2J_{V}+1)}\sum_{N}\frac{|\langle\psi_{V}||d||\psi_{N}\rangle|^{2}\times(\epsilon_{N}-\epsilon_{V})}{(E_{N}-E_{V})^{2}-\omega^{2}},$
(A.1)
$\alpha_{V}^{1}(\omega)=-\sqrt{\frac{6J_{V}}{(J_{V}+1)(2J_{V}+1)}}\sum_{N}(-1)^{J_{N}+J_{V}}\left\\{\begin{array}[]{ccc}J_{V}&1&J_{V}\\\
1&J_{N}&1\end{array}\right\\}\frac{|\langle\psi_{V}||d||\psi_{N}\rangle|^{2}\times
2\omega}{(E_{N}-E_{V})^{2}-\omega^{2}},$ (A.2)

$\alpha_{V}^{2}(\omega)=4\sqrt{\frac{5J_{V}(2J_{V}-1)}{6(J_{V}+1)(2J_{V}+1)(2J_{V}+3)}}\sum_{N}(-1)^{J_{N}+J_{V}}\left\\{\begin{array}[]{ccc}J_{V}&1&J_{N}\\\
1&J_{V}&2\end{array}\right\\}\frac{|\langle\psi_{V}||d||\psi_{N}\rangle|^{2}\times(\epsilon_{N}-\epsilon_{V})}{(E_{N}-E_{V})^{2}-\omega^{2}}.$
(A.3)

Here $J_{V}$ is the total angular momentum and $E_{V}$ is the ionization
potential of $\psi_{V}$. $\langle\psi_{V}||d||\psi_{N}\rangle$ is the reduced
matrix element of the electric dipole operator.

At a hyperfine level $F_{V}$ with nuclear spin $I$, the scalar component
($\alpha_{VF}^{0}(\omega)$) of the valence polarizability is equal to
$\alpha_{V}^{0}(\omega)$. This is because of the second-order scalar energy
shift does not depend on any hyperfine quantum number Beloy2009 . But the
hyperfine-induced vector ($\alpha_{VF}^{(1)}(\omega)$) and tensor
($\alpha_{VF}^{(2)}(\omega)$) components have different angular momentum
factors to be multiplied with the above expressions for
$\alpha^{1}_{V}(\omega)$ and $\alpha^{2}_{V}(\omega)$, respectively Beloy2009
; Dzuba2010 ; Das2020 :

$\alpha_{VF}^{(1)}(\omega)=(-1)^{J_{V}+F_{V}+I+1}\left\\{\begin{array}[]{ccc}F_{V}&J_{V}&I\\\
J_{V}&F_{V}&1\end{array}\right\\}\sqrt{\frac{F_{V}(2F_{V}+1)(2J_{V}+1)(J_{V}+1)}{J_{V}(F_{V}+1)}}\alpha_{V}^{(1)}(\omega)$
(A.4)

and

$\displaystyle\alpha_{VF}^{(2)}(\omega)=(-1)^{J_{V}+F_{V}+I}\left\\{\begin{array}[]{ccc}F_{V}&J_{V}&I\\\
J_{V}&F_{V}&2\end{array}\right\\}$
$\displaystyle\sqrt{\left(\frac{F_{V}(2F_{V}-1)(2F_{V}+1)}{(2F_{V}+3)(F_{V}+1)}\right)}$
(A.7) $\displaystyle\times$
$\displaystyle\sqrt{\left(\frac{(2J_{V}+3)(2J_{V}+1)(J_{V}+1)}{J_{V}(2J_{V}-1)}\right)}\alpha_{V}^{(2)}(\omega).$
(A.8)

The total valence polarizability at the hyperfine level
($\alpha_{F}^{V}(\omega)$) takes the form as

$\alpha_{F}^{V}(\omega)=C_{0}\alpha_{VF}^{0}(\omega)+C_{1}\alpha_{VF}^{1}(\omega)+C_{2}\alpha_{VF}^{2}(\omega),$
(A.9)

where $C_{i}$s are the coefficients mentioned in the main text.

###  IV.2 Details of numerical procedure:

Here, we present the numerical procedure to calculate the dynamic
polarizability of an atomic state discussed in the main text. The electron-
correlation significantly affects the valence polarizability of an atomic
state due to the loosely bound valence electron. Therefore, the precise
estimations of the scalar, vector, and tensor components of the valence
polarizability for an atomic state require correlation-exhaustive many-body
treatments with a sophisticated numerical approach.

### Calculation of the dynamic polarizability for 87Sr+ ion

To calculate the valence polarizabilities for $5S_{\frac{1}{2}}$,
$4D_{\frac{3}{2}}$, and $4D_{\frac{5}{2}}$ states, we consider all possible
non-negligible dipole matrix elements in Eq. (A.1) to (A.3). This sets the
highest principle quantum number of the running index $N$, which is considered
up to 25. According to the significance of each dipole matrix elements to the
summations in the equations, many-body calculations of different orders of
correlations are performed with negligible compromise in the accuracy of of
the polarizability values Bhowmik2018 . The most dominant dipole matrix
elements are associated with the intermediate states 5$P_{1/2,3/2}$ to
8$P_{1/2,3/2}$ and 4$F_{5/2,7/2}$ to 6$F_{5/2,7/2}$ as $\psi_{N}$, and they
are computed using a relativistic coupled-cluster theory having cluster
operators containing single, double, and valence triple excitations in linear
and non-linear forms Lindgren1985 ; Bishop1987 ; Lindgren1986 ; Lindgren_1985
; Bartlett2007 ; Dutta2016 ; Bhowmik2017 ; Das2018 ; Biswas2018 . Relatively
less significant dipole matrix elements are involved with the states
9$P_{1/2,3/2}$ to 12$P_{1/2,3/2}$ and 7$F_{5/2,7/2}$ to 12$F_{5/2,7/2}$ are
evaluated using the 2nd-order relativistic many-body perturbation theory
Lindgren1986 . The remaining matrix elements contribute little to the
summations of Eq. (A.1) to (A.3) and are calculated using the Dirac-Fock
method. Also, to achieve a better accuracy in the total dynamic polarizability
values, we have utilized the transition energies from the experimental data
NIST . By employing the above methods, the static scalar polarizabilities
$(\alpha_{V}^{0}(0))$ of $5S_{\frac{1}{2}}$, $4D_{\frac{3}{2}}$, and
$4D_{\frac{5}{2}}$ states are calculated as 87.68 a.u., 55.92 a.u., and 56.21
a.u., respectively, and static tensor polarizabilities $(\alpha_{V}^{2}(0))$
of $4D_{\frac{3}{2}}$ and $4D_{\frac{5}{2}}$ states are computed as $-34.67$
a.u. and $-47.12$ a.u., respectively Bhowmik2018 .

The core polarizability $\alpha^{C}(\omega)$ is valence-state independent
quantity and it can be calculated quite accurately for a system having noble
gas electronic configuration with core polarization corrected dipole matrix
elements using 3rd-order relativistic many-body perturbation theory
Johnson1996 ; Dutta2020 . Our calculation yields that the static core POL
($\alpha^{C}(0)$) of the ion is 6.103 a.u.. The static core-valence parts
$\alpha^{VC}(0)$ for the states $5S_{\frac{1}{2}}$, $4D_{\frac{3}{2}}$, and
$4D_{\frac{5}{2}}$ are calculated to be $-0.25$ a.u., $-0.38$ a.u., and
$-0.42$ a.u., respectively. All the magic and tune-out wavelengths calculated
for the ion using the above mentioned numerical procedure have maximum
uncertainty of around $\pm 1\%$. To calculate the uncertainty, we replace the
most important dipole matrix elements of the transitions:
$5S_{\frac{1}{2}}\rightarrow 5P_{\frac{1}{2},\frac{3}{2}}$ for
$5S_{\frac{1}{2}}$, $4D_{\frac{3}{2}}\rightarrow
5P_{\frac{1}{2},\frac{3}{2}},4F_{\frac{5}{2}}$ for $4D_{\frac{3}{2}}$, and
$4D_{\frac{5}{2}}\rightarrow 5P_{\frac{3}{2}},4F_{\frac{5}{2},\frac{7}{2}}$
for $4D_{\frac{5}{2}}$ by the corresponding matrix elements calculated by
Safronova Safronova2010  using single-double all-order method including
partial triple excitations and recalculate the magic and tune-out wavelengths.
The maximum difference between these recalculated wavelengths and the
corresponding wavelengths obtained from our method gives the uncertainty of
our results.

### Calculation of the dynamic polarizability for 87Rb atom

To determine the valence polarizability of the ground state $5S_{\frac{1}{2}}$
of 87Rb atom at the hyperfine level, Eq. (A.9) reduces to
$\alpha_{F}^{V}(\omega)=C_{0}\alpha_{VF}^{0}(\omega)+C_{1}\alpha_{VF}^{1}(\omega)$.
The tensor part of the polarizability is zero for $5S_{\frac{1}{2}}$ state.
Similar to 87Sr+ ion, here also, the dipole matrix elements involved with the
states 5$P_{1/2,3/2}$ to 8$P_{1/2,3/2}$ and 9$P_{1/2,3/2}$ to 12$P_{1/2,3/2}$
are computed using the relativistic coupled-cluster theory and 2nd-order
relativistic many-body perturbation theory, respectively. 13$P_{1/2,3/2}$ to
25$P_{1/2,3/2}$ states and their associated matrix elements with
$5S_{\frac{1}{2}}$ state are calculated using the Dirac-Fock method. To
achieve better accuracy, the energy values of the states, used to calculate
the polarizability values, are extracted from the experimental measurements
NIST . For $5S_{\frac{1}{2}}$ state of 87Rb atom, we find that the static
scalar, core, and core-valence polarizability values are 314.14 a.u., 9.11
a.u., and $-0.26$ a.u., respectively. The tune-out wavelengths of the
hyperfine spin states of 87Rb atom, discussed in the main text, have maximum
uncertainty of $\pm 0.1\%$.

### Zeeman shift for 87Rb and 23Na atoms

The Zeeman shifts for 87Rb and 23Na atoms are evaluated using
$\hat{H}_{Z}=-\beta\hat{F}_{Z}+\gamma\hat{F}_{Z}^{2}$, where $\beta$ and
$\gamma$ are the linear and quadratic Zeeman shifts, respectively.
$\hat{F}_{Z}$ represents the $z$-component of the external magnetic field. The
coefficients $\beta$ and $\gamma$ can be obtained from the power series
expansion of the Breit-Rabi formula Vanier1988 . The fine structure ($g_{J}$)
and nuclear Landé $g$-factors ($g_{I}$), used in the Breit-Rabi formula, are
2.00233113 and $-0.0009951414$, respectively, for 87Rb atom, while they are
2.0022960 and $-0.000804610$, respectively, for 23Na atom Arimondo1977 . Also,
the ground state hyperfine splittings of 87Rb and 23Na atoms, require in the
Breit-Rabi formula, have the values 6.8 GHz and 1.7 GHz Arimondo1977 ,
respectively.

###  IV.3 List of magic wavelengths with the corresponding vector and total
polarizabilities

A list of magic wavelengths and the corresponding total polarizability values
at these wavelengths for the transitions $5S_{\frac{1}{2}}\rightarrow
4D_{\frac{3}{2},\frac{5}{2}}$ are presented in Table LABEL:I-Table LABEL:III.
Moreover, at each magic wavelength, values of the vector polarizabilities of
the relevant states are quoted. Results are displayed for the OAM=+1 and +2,
while the focusing angles of the beam are considered as 50∘, 60∘ and 70∘. As
seen in the tables, all the transitions between the magnetic sublevels of
$5S_{\frac{1}{2}}$ and $4D_{\frac{3}{2},\frac{5}{2}}$ states produce five sets
of magic wavelengths except few cases, say, $5S_{\frac{1}{2}}\rightarrow
4D_{\frac{5}{2}}$ transition for OAM=+2. Depending on the proximity of the
resonances, the strength of the vector polarizability of one of the state
dominates over the other state for a particular value of magic wavelength. For
$5S_{\frac{1}{2}}\rightarrow 4D_{\frac{5}{2}}(+5/2)$ transition, we have found
(in table III ) two sets of infrared magic wavelengths are missing at the
focusing angle 50∘ and 60∘ of the beam, but all five sets of magic wavelengths
present at 70∘, when the projected beam has OAM=+2. This highlights the direct
effect of spin-orbit coupling of linearly polarized vortex beam on the magic
wavelengths. All the tables show that the magic wavelengths fall in the
visible and ultra-violet region of the electromagnetic spectrum support the
blue-detuned trapping scheme confining the ion in the low intensity region of
the beam Kennedy2014 ; Kuga1997 ; Wright2000 ; Arlt2001 .

Table 1: Magic wavelengths (in nm) with corresponding polarizabilities (in a.u.) of 87Sr+ for different focusing angles of the linearly polarized vortex beam (with OAM=+1 and +2) for transitions $5S_{\frac{1}{2}}(+1/2)$ $\rightarrow$ $4D_{\frac{3}{2}}(M_{J})$. The values in the parentheses refer vector polarizabilities at the corresponding magic wavelengths. In the parenthesis (a, b), a and b indicate the vector polarizabilities for $5S_{\frac{1}{2}}(+1/2)$ and $4D_{\frac{3}{2}}(M_{J})$ states, respectively, in a.u.. State | $\lambda^{50^{\circ}}$ | $\alpha^{50^{\circ}}$ | $\lambda^{60^{\circ}}$ | $\alpha^{60^{\circ}}$ | $\lambda^{70^{\circ}}$ | $\alpha^{70^{\circ}}$
---|---|---|---|---|---|---
($4D_{\frac{3}{2}}(M_{J})$) | OAM=+1
$(+\frac{1}{2})$ | 1627.26 | 113.53 (-0.16, -7.20) | 1598.71 | 115.45 (-0.16, -7.81) | 1571.15 | 117.37 (-0.16, -7.27)
| 992.67 | 126.99 (-0.33, 12.91) | 981.97 | 129.94 (-0.34, 13.95) | 981.97 | 132.77 (-0.32, 13.14)
| 403.57 | 17.32 (158.00, 3.10) | 403.57 | 17.32 (156.69, 3.07) | 403.57 | 17.32 (158.72, 2.91)
| 213.21 | -29.88 (-0.13, -0.93) | 213.21 | -30.79 (-0.13, -0.84) | 213.21 | -31.81 (-0.13, -0.24)
| 200.01 | -24.11 (-0.10, -1.37) | 200.45 | -25.01 (-0.10, -1.43) | 200.45 | -26.03 (-0.09, -1.24)
$(-\frac{1}{2})$ | 1786.80 | 111.60 (-0.15, 5.20) | 1752.44 | 114.54 (-0.15, 6.21) | 1752.44 | 116.47 (-0.14, 5.52)
| 1003.60 | 126.99 (-0.33, -12.51) | 992.67 | 129.94 (-0.34, -13.25) | 992.67 | 131.86 (-0.32, -12.93)
| 403.57 | 10.53 (158.00, -3.10) | 403.57 | 10.53 (156.69, -3.07) | 403.57 | 11.54 (158.72, -2.91)
| 213.21 | -29.88 (-0.13, 0.93) | 213.21 | -30.79 (-0.13, 0.84) | 213.21 | -31.81 (-0.13, 0.24)
| 200.45 | -25.01 (-0.10, 1.47) | 200.90 | -25.01 (-0.10, 1.43) | 201.34 | -26.03 (-0.09, 1.64)
$(+\frac{3}{2})$ | 1368.27 | 116.47 (-0.19, -40.15) | 1389.13 | 118.39 (-0.20, -37.93) | 1406.28 | 120.32 (-0.19, -34.55)
| 961.25 | 128.92 (-0.35, 43.73) | 961.25 | 130.84 (-0.37, 43.90) | 961.25 | 133.79 (-0.34, 42.50)
| 403.57 | 23.09 (158.00, 9.25) | 403.57 | 23.09 (156.69, 9.26) | 403.57 | 23.09 (158.72, 8.75)
| 213.21 | -29.88 (-0.13, -2.78) | 212.71 | -30.79 (-0.13, -18.52) | 213.21 | -31.81 (-0.13, -2.64)
| 199.58 | -24.11 (-0.10, -4.33) | 199.58 | -25.01 (-0.10, -5.23) | 199.58 | -25.01 (-0.08, -3.46)
$(-\frac{3}{2})$ | 1815.27 | 111.60 (-0.15, 15.61) | 1815.27 | 113.53 (-0.21, 15.65) | 1815.27 | 116.47 (-0.13, 14.78)
| 981.97 | 128.01 (-0.34, -41.81) | 992.67 | 128.92 (-0.34, -40.55) | 992.67 | 131.86 (-0.32, -38.00)
| 403.57 | 4.75 (158.00, -9.25) | 403.57 | 4.75 (156.69, -9.26) | 403.57 | 5.77 (158.72, -8.75)
| 213.21 | -29.88 (-0.13, 2.78) | 213.21 | -30.79 (-0.13, 2.80) | 213.21 | -31.81 (-0.13, 2.64)
| 202.14 | -25.01 (-0.10, 6.21) | 202.14 | -26.03 (-0.10, 6.13) | 201.79 | -26.03 (-0.09, 5.29)
($4D_{\frac{3}{2}}(M_{J})$) | OAM=+2
$(+\frac{1}{2})$ | 2312.86 | 103.00 (-0.05, -1.66) | 2201.13 | 104.92 (-0.05, -1.91) | 2052.40 | 107.75 (-0.06, -2.15)
| 1047.43 | 117.37 (-0.15, -5.16) | 1035.53 | 119.30 (-0.16, -1.46) | 1035.53 | 122.24 (-0.16, -1.43)
| 405.73 | 13.47 (115.64, 1.51) | 403.57 | 14.37 (86.72, 1.59) | 405.73 | 14.37 (119.91, 1.56)
| 213.21 | -27.96 (-0.06, -0.55) | 213.21 | -28.86 (-0.07, -0.48) | 213.21 | -28.86 (-0.06, -0.47)
| 199.66 | -22.18 (-0.05, -0.71) | 200.01 | -23.09 (-0.05, -0.74) | 199.58 | -23.09 (-0.05, -0.61)
$(-\frac{1}{2})$ | 2559.74 | 101.98 (-0.04, 1.43) | 2360.80 | 104.92 (-0.05, 1.71) | 2201.13 | 107.75 (-0.06, 1.88)
| 1035.53 | 117.37 (-0.16, 1.38) | 1035.53 | 119.30 (-0.16, 1.46) | 1035.53 | 122.24 (-0.16, 1.43)
| 403.57 | 10.53 (82.11, -1.51) | 403.57 | 10.53 (86.72, -1.59) | 405.73 | 11.54 (119.91, -1.56)
| 213.21 | -27.96 (-0.06, 0.55) | 213.21 | -28.86 (-0.07, 0.48) | 213.21 | -28.86 (-0.06, 0.47)
| 199.58 | -22.18 (-0.05, 0.39) | 200.01 | -23.10 (-0.05, 0.74) | 200.01 | -23.09 (-0.05, 0.69)
$(+\frac{3}{2})$ | 1276.28 | 111.60 (-0.11, -33.13) | 1276.28 | 112.62 (-0.11, -35.00) | 1290.75 | 115.45 (-0.11, -28.42)
| 943.34 | 122.24 (-0.18, 22.70) | 943.34 | 124.17 (-0.20, 23.98) | 943.34 | 126.99 (-0.19, 23.54)
| 405.73 | 19.24 (115.64, 4.52) | 405.73 | 19.24 (122.13, 4.79) | 405.73 | 19.24 (119.91, 4.70)
| 213.21 | -27.96 (-0.06, -1.37) | 213.21 | -28.86 (-0.07, -1.45) | 213.21 | -28.86 (-0.06, -1.42)
| 200.90 | -22.18 (-0.05, -2.36) | 200.90 | -23.09 (-0.05, -2.50) | 200.45 | -23.09 (-0.05, -2.25)
$(-\frac{3}{2})$ | 1428.32 | 108.77 (-0.09, 16.18) | 1451.06 | 109.68 (-0.09, 16.10) | 1474.54 | 112.62 (-0.09, 14.67)
| 951.22 | 121.22 (-0.18, -22.19) | 951.22 | 123.15 (-0.19, -23.54) | 951.22 | 126.99 (-0.19, -23.01)
| 403.57 | 9.62 (82.11, -4.51) | 403.57 | 9.62 (86.72, -4.75) | 405.73 | 9.62 (119.91, -4.70)
| 213.21 | -27.96 (-0.06, 1.37) | 213.21 | -28.86 (-0.07, 1.45) | 213.21 | -28.86 (-0.06, 1.42)
| 202.14 | -23.09 (-0.05, 2.90) | 202.14 | -24.11 (-0.05, 3.06) | 201.79 | -24.11 (-0.05, 2.84)
Table 2: Magic wavelengths (in nm) with corresponding polarizabilities (in a.u.) of 87Sr+ for different focusing angles of linearly polarized vortex beam with OAM=+1 for transitions $5S_{\frac{1}{2}}(+1/2)$ $\rightarrow$ $4D_{\frac{5}{2}}(M_{J})$. The values in the parentheses refer vector polarizabilities at the corresponding magic wavelengths. In the parenthesis (a, b), a and b indicate the vector polarizabilities for $5S_{\frac{1}{2}}(+1/2)$ and $4D_{\frac{5}{2}}(M_{J})$ states, respectively, in a.u.. State | $\lambda^{50^{\circ}}$ | $\alpha^{50^{\circ}}$ | $\lambda^{60^{\circ}}$ | $\alpha^{60^{\circ}}$ | $\lambda^{70^{\circ}}$ | $\alpha^{70^{\circ}}$
---|---|---|---|---|---|---
($4D_{\frac{5}{2}}(M_{J})$) | OAM=+1
$(+\frac{1}{2})$ | 1544.52 | 114.54 (-0.17, -4.39) | 1523.86 | 116.47 (-0.18, -4.56) | 1523.86 | 118.39 (-0.17, -4.31)
| 1122.25 | 122.24 (-0.28, -5.30) | 1122.25 | 124.17 (-0.28, -5.33) | 1122.25 | 126.99 (-0.26, -5.46)
| 403.57 | 18.22 (158.00, 2.14) | 403.57 | 18.22 (156.69, 2.15) | 403.57 | 18.22 (158.72, 2.03)
| 212.22 | -29.88 (-0.13, 1.44) | 212.22 | -29.88 (-0.13, 1.45) | 212.22 | -30.79 (-0.12, 1.36)
| 202.14 | -25.01 (-0.10, -0.89) | 202.59 | -26.03 (-0.10, -1.03) | 202.14 | -26.03 (-0.10, -0.91)
$(-\frac{1}{2})$ | 1656.85 | 113.53 (-0.16, 3.77) | 1656.85 | 114.54 (-0.16, 3.78) | 1627.26 | 117.37 (-0.15, 3.72)
| 1122.25 | 122.24 (-0.28, 5.30) | 1122.25 | 124.17 (-0.28, 5.33) | 1122.25 | 126.99 (-0.26, 5.46)
| 403.57 | 13.47 (158.00, -2.14) | 403.57 | 13.47 (156.69, -2.15) | 403.57 | 14.37 (158.72, -2.03)
| 212.22 | -29.88 (-0.13, -1.44) | 212.22 | -30.79 (-0.13, -1.45) | 212.22 | -30.79 (-0.12, -1.36)
| 202.59 | -25.01 (-0.10, 1.02) | 202.59 | -26.03 (-0.10, 1.03) | 202.59 | -26.94 (-0.10, 0.96)
$(+\frac{3}{2})$ | 1406.28 | 115.45 (-0.20, -18.64) | 1428.32 | 117.37 (-0.20, -17.34) | 1428.32 | 120.32 (-0.19, -16.37)
| 1122.25 | 122.24 (-0.28 -15.94) | 1122.25 | 124.17 (-0.28, -15.99) | 1122.25 | 126.99 (-0.26, -15.09)
| 403.57 | 23.09 (158.00, 6.52) | 403.57 | 22.07 (156.69, 6.44) | 403.57 | 22.07 (158.72, 6.10)
| 212.22 | -28.86 (-0.13, 4.33) | 212.22 | -30.79 (-0.13, 4.34) | 212.22 | -30.79 (-0.12, 4.10)
| 201.79 | -25.01 (-0.10, -2.68) | 201.79 | -25.01 (-0.10, -2.69) | 201.79 | -26.03 (-0.10, -2.56)
$(-\frac{3}{2})$ | 1712.91 | 112.62 (-0.15, 10.46) | 1687.53 | 114.54 (-0.16, 10.86) | 1687.53 | 117.37 (-0.15, 10.26)
| 1122.25 | 122.24 (-0.28, 15.94) | 1122.25 | 124.17 (-0.28, 15.99) | 1122.25 | 126.99 (-0.26, 15.09)
| 403.57 | 9.62 (158.00, -6.52) | 403.57 | 9.62 (156.69, -6.44) | 403.57 | 9.62 (158.72, -6.10)
| 212.22 | -29.88 (-0.13, -4.33) | 212.22 | -30.79 (-0.13, -4.34) | 212.22 | -30.79 (-0.12, -4.10)
| 203.05 | -26.03 (-0.10, 3.59) | 203.50 | -26.94 (-0.10, 4.04) | 203.50 | -26.94 (-0.10, 3.82)
$(+\frac{5}{2})$ | 1276.28 | 119.30 (-0.22, -46.32) | 1276.28 | 120.32 (-0.22, -46.44) | 1309.29 | 123.15 (-0.21, -36.17)
| 1122.25 | 122.24 (-0.28, -26.57) | 1122.25 | 124.17 (-0.28, -26.64) | 1122.25 | 126.99 (-0.26, -25.16)
| 403.57 | 26.93 (158.00, 10.70) | 403.57 | 26.94 (156.69, 10.73 ) | 403.57 | 26.94 (158.72, 10.15)
| 212.22 | -28.86 (-0.13, 7.23) | 212.22 | -30.79 (-0.13, 7.24) | 212.22 | -30.79 (-0.12, 6.84)
| 201.79 | -25.01 (-0.10, -4.48) | 201.79 | -26.03 (-0.10, -4.49) | 201.79 | -26.03 (-0.10, -4.24)
$(-\frac{5}{2})$ | 1687.53 | 112.62 (-0.16, 18.04) | 1712.91 | 114.54 (-0.15, 17.48) | 1712.91 | 117.37 (-0.14, 16.51)
| 1122.25 | 122.24 (-0.28, 26.57) | 1122.25 | 124.17 (-0.28, 26.64) | 1122.25 | 126.99 (-0.26, 25.16)
| 403.57 | 5.77 (158.00, -10.70) | 403.57 | 5.77 (156.69, -10.73) | 403.57 | 6.68 (158.72, -10.15)
| 212.22 | -29.88 (-0.13, -7.23) | 212.22 | -29.88 (-0.13, -7.24) | 212.22 | -30.79 (-0.12, -6.84)
| 204.41 | -26.03 (-0.10, 8.15) | 203.95 | -26.03 (-0.10, 7.46) | 203.95 | -26.93 (-0.10, 7.03)
Table 3: Magic wavelengths (in nm) with corresponding polarizabilities (in a.u.) of 87Sr+ for different focusing angles of the linearly polarized vortex beam with OAM=+2 for transitions $5S_{\frac{1}{2}}(+1/2)$ $\rightarrow$ $4D_{\frac{5}{2}}(M_{J})$. The values in the parentheses refer vector polarizabilities at the corresponding magic wavelengths. In the parenthesis (a, b), a and b indicate the vector polarizabilities for $5S_{\frac{1}{2}}(+1/2)$ and $4D_{\frac{5}{2}}(M_{J})$ states, respectively, in a.u.. State | $\lambda^{50^{\circ}}$ | $\alpha^{50^{\circ}}$ | $\lambda^{60^{\circ}}$ | $\alpha^{60^{\circ}}$ | $\lambda^{70^{\circ}}$ | $\alpha^{70^{\circ}}$
---|---|---|---|---|---|---
($4D_{\frac{5}{2}}(M_{J})$) | OAM=+2
$(+\frac{1}{2})$ | 2360.80 | 103.00 (-0.05, -0.93) | 2149.21 | 105.83 (-0.05, -1.17) | 2052.40 | 107.75 (-0.06, -1.25)
| 1122.25 | 114.54 (-0.14, -2.61) | 1122.25 | 116.47 (-0.14, -2.75) | 1122.25 | 119.30 (-0.14, -2.70)
| 405.73 | 14.37 (115.64, 1.05) | 405.73 | 15.39 (122.13, 1.11) | 405.73 | 15.39 (119.91, 1.09)
| 212.22 | -26.94 (-0.06, 0.71) | 212.22 | -27.96 (-0.07, 0.73) | 212.22 | -28.86 (-0.06, 0.73)
| 201.34 | -23.09 (-0.05, -0.40) | 201.79 | -23.09 (-0.05, -0.44) | 201.34 | -24.11 (-0.05, -0.42)
$(-\frac{1}{2})$ | 2489.80 | 103.00 (-0.05, 0.87) | 2278.17 | 104.81 (-0.05, 1.05) | 2149.21 | 107.75 (-0.06, 1.15)
| 1122.25 | 114.54 (-0.14, 2.61) | 1122.25 | 116.47 (-0.14, 2.75) | 1122.25 | 119.30 (-0.14, 2.70)
| 405.73 | 12.45 (115.64, -1.05) | 403.57 | 13.24 (86.72, -1.11) | 405.73 | 13.47 (119.91, -1.09)
| 212.22 | -26.94 (-0.06, -0.71) | 212.02 | -27.39 (-0.07, 2.09) | 212.22 | -28.86 (-0.06, -0.73)
| 201.79 | -23.09 (-0.05, 0.44) | 201.88 | -23.66 (-0.05, 0.44) | 201.79 | -24.11 (-0.05, 0.46)
$(+\frac{3}{2})$ | 1544.52 | 107.75 (-0.09, -6.45) | 1544.52 | 108.77 (-0.09, -6.82) | 1523.86 | 111.60 (-0.09, -6.93)
| 1122.25 | 114.54 (-0.14, -7.81) | 1122.25 | 116.47 (-0.14, -8.25) | 1122.25 | 119.30 (-0.14, -8.10)
| 405.73 | 18.22 (115.64, 3.15) | 405.73 | 18.22 (122.13, 3.34) | 405.73 | 18.22 (119.91, 3.27)
| 212.22 | -26.94 (-0.06, 2.13) | 212.22 | -27.96 (-0.07, 2.25) | 212.22 | -28.86 (-0.06, 2.20)
| 201.79 | -23.09 (-0.05, -1.31) | 201.79 | -23.09 (-0.05, -1.38) | 201.79 | -24.11 (-0.05, -1.37)
$(-\frac{3}{2})$ | 1752.44 | 105.83 (-0.07, 4.85) | 1712.91 | 107.75 (-0.07, 5.41) | 1712.91 | 109.68 (-0.07, 5.31)
| 1122.25 | 114.54 (-0.14, 7.81) | 1122.25 | 116.47 (-0.14, 8.25) | 1122.25 | 119.30 (-0.14, 8.10)
| 405.73 | 11.54 (115.64, -3.15) | 403.57 | 11.54 (86.72, -3.34) | 405.73 | 11.54 (119.91, -3.27)
| 212.22 | -26.94 (-0.06, -2.13) | 212.22 | -27.96 (-0.07, -2.25) | 212.22 | -28.86 (-0.06, -2.20)
| 202.59 | -23.09 (-0.05, 1.50) | 202.59 | -24.11 (-0.05, 1.59) | 202.59 | -24.11 (-0.05, 1.56)
$(+\frac{5}{2})$ |  |  |  |  | 1192.76 | 117.37 (-0.13, -37.85)
|  |  |  |  | 1136.24 | 119.29 (-0.14, -41.59)
| 405.73 | 22.07 (115.64, 5.25) | 405.73 | 22.07 (122.13, 5.55) | 405.73 | 22.07 (119.91, 5.45)
| 212.22 | -26.94 (-0.06, 3.54) | 212.22 | -27.96 (-0.07, 3.73) | 212.22 | -28.86 (-0.06, 3.67)
| 203.05 | -23.09 (-0.05, -2.72) | 202.59 | -24.11 (-0.05, -2.65) | 202.59 | -24.11 (-0.05, -2.60)
$(-\frac{5}{2})$ | 1276.28 | 110.70 (-0.11, 22.70) | 1290.75 | 112.62 (-0.11, 21.72) | 1328.38 | 114.54 (-0.11, 18.54)
| 1122.25 | 114.54 (-0.14, 13.02) | 1122.25 | 116.47 (-0.14, 13.74) | 1122.25 | 119.30 (-0.14, 13.50)
| 405.73 | 11.54 (115.64, -5.25) | 403.57 | 11.54 (86.72, -5.55) | 405.73 | 11.54 (119.91, -5.45)
| 212.22 | -26.94 (-0.06, -3.54) | 212.22 | -27.96 (-0.07, -3.73) | 212.22 | -28.86 (-0.06, -3.67)
| 204.78 | -24.11 (-0.05, 4.29) | 204.41 | -24.11 (-0.05, 4.23) | 203.95 | -25.01 (-0.05, 3.77)

## References

  * (1) C. Champenois, M. Houssin, C. Lisowski,M. Knoop, G. Hagel, M. Vedel, and F. Vedel, Phys. Lett. A 331, 298 (2004).
  * (2) C. W. Chou, D. B. Hume, J. C. J. Koelemeij, D. J. Wineland, and T. Rosenband, Phys. Rev. Lett. 104, 070802 (2010).
  * (3) A. D. Ludlow, M. M. Boyd, J. Ye, E. Peik, and P. O. Schmidt, Rev. Mod. Phys. 87, 637 (2015).
  * (4) T. Lauprêtre, R. B. Linnet, I. D. Leroux, H. Landa, A. Dantan, and M. Drewsen, Phys. Rev. A 99, 031401(R) (2019).
  * (5) J. Schmidt, A. Lambrecht, P. Weckesser, M. Debatin, L. Karpa, and T. Schaetz, Phys. Rev. X 8, 021028 (2018).
  * (6) A. Lambrecht, J. Schmidt, P. Weckesser, M. Debatin, L. Karpa, and T. Schaetz, Nat. Photonics 11, 704 (2017).
  * (7) T. Schaetz, J. Phys. B: At. Mol. Opt. Phys. 50, 102001 (2017).
  * (8) G. W. Biedermann, X. Wu, L. Deslauriers, S. Roy, C. Mahadeswaraswamy, and M. A. Kasevich, Phys. Rev. A 91, 033629 (2015).
  * (9) H. S. Margolis, J. Phys. B 42, 154017 (2009).
  * (10) T. Nicholson, S. Campbell, R. Hutson, G. Marti, B. Bloom, R. McNally, W. Zhang, M. Barrett, M. Safronova, G. Strouse et al., Nat. Commun. 6, 6896 (2015).
  * (11) P. Rosenbusch, S. Ghezali, V. A. Dzuba, V. V. Flambaum, K. Beloy, and A. Derevianko, Phys. Rev. A 79, 013404 (2009).
  * (12) R. C. Dong, R. Wei, Y. B. Du, F. Zou, J. D. Lin, and Y. Z. Wang, Appl. Phys. Lett. 106, 152402 (2015).
  * (13) B. Arora, M. S. Safronova, and C. W. Clark, Phys. Rev. A 76, 052509 (2007).
  * (14) C. Lacroûte, K. S. Choi ,A. Goban, D. J. Alton, D. Ding, N. P. Stern, H. J. Kimble, New J. Phys. 14, 023056 (2012).
  * (15) B. Arora and B. K. Sahoo, Phys. Rev. A 86, 033416 (2012).
  * (16) V. V. Flambaum, V. A. Dzuba, and A. Derevianko, Phys. Rev. Lett. 101, 220801 (2008).
  * (17) A. Bhowmik, N. N. Dutta, and S. Majumder, Phys. Rev. A 97, 022511 (2018).
  * (18) A. Das, A. Bhowmik, N. N. Dutta, and S. Majumder, Phys. Rev. A 102, 012801 (2020).
  * (19) H Kim, H S Han, and D. Cho, Phys. Rev. Letts., 111, 243004 (2013)
  * (20) L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, Phys. Rev. A 45, 8185 (1992).
  * (21) P. K. Mondal, B. Deb, S. Majumder, Phys. Rev. A 89, 063418 (2014).
  * (22) A. Bhowmik, P. K. Mondal, S. Majumder, and B. Deb, Phys. Rev. A 93, 063852 (2016).
  * (23) C. T. Schmiegelow, J. Schulz, H. Kaufmann, T. Ruster, U. G. Poschinger, and F. Schmidt-Kaler, Nature Communications 7, 12998 (2016).
  * (24) A. Bhowmik and S. Majumder, J. Phys. Commun. 2, 125001 (2018).
  * (25) M. Babiker, C. R. Bennett, D. L. Andrews, and L. C. D. Romero, Phys. Rev. Lett. 89, 143601 (2002).
  * (26) V. E. Lembessis, D. Ellinas, and M. Babiker, Phys. Rev. A 84, 043422 (2011).
  * (27) A. Afanasev, C. E. Carlson, and A. Mukherjee, Phys. Rev. A 88, 033841 (2013).
  * (28) J. Romero, D. Giovannini, S. Franke-Arnold, S. M. Barnett, and M. J. Padgett, Phys. Rev. A 86, 012334 (2012).
  * (29) S. Das, A. Bhowmik, K. Mukherjee, S. Majumder, J. Phys. B: At. Mol. Opt. Phys. 53, 025302 (2020).
  * (30) G. F. Quinteiro, C. T. Schmiegelow, F. Schmidt-Kaler, arXiv:2004.00040
  * (31) F. L. Kien, P. Schneeweiss, and A. Rauschenbeute, Eur. Phys. J. D 67 92 (2013).
  * (32) B. Albrecht, Y. Meng, C. Clausen, A. Dareau, P. Schneeweiss, and A. Rauschenbeutel, Phys. Rev. A, 94, 061401(R) (2016).
  * (33) P. Schneeweiss, F. L. Kien, and A. Rauschenbeutel, New J. Phys., 16, 013014 (2014).
  * (34) J. Stenger, S. Inouye, D. M. Stampe-Kurn, H. J. Miesner, A. P. Chikkatur, and W. Ketterle, Nature 396, 345 (1998).
  * (35) M. -S. Chang, C. D. Hamley, M. D. Barrett, J. A. Sauer, K. M. Fortier, W. Zhang, L. You, and M. S. Chapman, Phys. Rev. Lett. 92, 140403 (2004).
  * (36) M. -S. Chang, Q. S. Qin, W. X. Zhang, L. You, and M. S. Chapman, Nat. Phys. 1, 111 (2005).
  * (37) C. Klempt, O. Topic, G. Gebreyesus, M. Scherer, T. Henninger, P. Hyllus, W. Ertmer, L. Santos, and J. J. Arlt, Phys. Rev. Lett. 103, 195302 (2009).
  * (38) D. M. Stamper-Kurn and M. Ueda, Rev. Mod. Phys. 85, 1191 (2013).
  * (39) N. Akerman, N. Navon, S. Kotler, Y. Glickman, and R. Ozeri, New J. of Phys. 17, 113060 (2015).
  * (40) Y. Shapira, R. Shaniv, T. Manovitz, N. Akerman, and R. Ozeri, Phys. Rev. Lett. 121, 180502 (2018).
  * (41) L. J. LeBlanc and J. H. Thywissen, Phys. Rev. A 75,053612 (2007).
  * (42) X. Wang, J. Jiang, L. -Y. Xie, D. -H. Zhang, and C. -Z. Dong Phys. Rev. A 94, 052510 (2016).
  * (43) J. Mitroy, M. S. Safronova, and C. W. Clark, J. Phys. B: At. Mol. Opt. Phys. 43, 202001 (2010).
  * (44) N. N. Dutta, S. Roy, and P. C. Deshmukh, Phys. Rev. A 92, 052510 (2015).
  * (45) T. K. Ghosh, A. K. Das, M. Castro, S. Canuto, and P. K. Mukherjee, Phys. Rev. A 48, 2686 (1993).
  * (46) N. N. Dutta, Chem. Phys. Lett. 758, 137911 (2020).
  * (47) B. Richards and E. Wolf, Proc. R. Soc. London, Ser. A 253, 358 (1959).
  * (48) A. Boivin and E. Wolf, Phys. Rev. 138, B1561 (1965).
  * (49) Y. Zhao, J. S. Edgar, G. D. M. Jeffries, D. McGloin, and D. T. Chiu , Phys. Rev. Lett. 99, 073901 (2007).
  * (50) I. Lindgren and J. Morrison, Atomic Many-body Theory, ed. G. E. Lambropoulos and H. Walther(3rd ed.; Berlin: Springer),3 (1985).
  * (51) R. F. Bishop, and H. G. K?ummel, Physics Today March 40, 52 (1987).
  * (52) I. Lindgren, and J. Morrison, Atomic Many-Body Theory, (Springer, 1986).
  * (53) I. Lindgren, Phys. Rev. A 31, 1273 (1985).
  * (54) R. J. Bartlett and M. Musial, Reviews of Modern Physics 79, 291 (2007).
  * (55) N. N. Dutta and S. Majumder, Indian Journal of Physics 90, 373 (2016).
  * (56) A. Bhowmik, N. N. Dutta, and S. Roy, Astrophys. J.836, 125 (2017).
  * (57) A. Bhowmik, S. Roy, N. N. Dutta, and S. Majumder, J. Phys. B: At. Mol. Opt. Phys. 50, 125005 (2017).
  * (58) A. Das, A. Bhowmik, N. N. Dutta, and S. Majumder, J. Phys. B: At. Mol. Opt. Phys. 51, 025001 (2018).
  * (59) J. Vanier and C. Audoin, The Quantum Physics of Atomic Frequency Standards (Hilger, Philadelphia, 1988).
  * (60) W. X. Zhang, D. L. Zhou, M. S. Chang, M. S. Chapman, and L. You, Phys. Rev. A 72, 013602 (2005).
  * (61) X. Li, B. Zhu, X. He, F. Wang, M. Guo, Z. Xu, S. Zhang, and D. Wang, Phys. Rev. Lett. 114, 255301 (2015).
  * (62) F. Fang, S. Wu, A. Smull, J. A. Isaacs, Y. Wang, C. H. Greene, and D. M. Stamper-Kurn, Phys. Rev. A 101, 012703 (2019).
  * (63) Q. -Q. Hu, C. Freier, Y. Sun, B. Leykauf, V. Schkolnik, J. Yang, M. Krutzik, and A. Peters, Phys. Rev. A 97, 013424 (2018).
  * (64) F. Schmidt, D. Mayer, M. Hohmann, T. Lausch, F. Kindermann, and A. Widera1, Phys. Rev. A 93, 022507 (2016).
  * (65) R. Trubko, M. D. Gregoire, W. F. Holmgren, and A. D. Cronin, Phys. Rev. A 95, 052507 (2017).
  * (66) L. Li, B. Zhu, B. Lu, S. Zhang, and D. Wang, Phys. Rev. A 101, 053611 (2020).
  * (67) Y. Shi and Q. Niu, Phys. Rev. Lett. 96, 140401 (2006).
  * (68) K. Beloy, Theory of the ac stark effect on the atomic hyperfine structure and applications to microwave atomic clocks, Doctoral dissertation, University of Nevada, Reno, USA, 2009.
  * (69) V. A. Dzuba, V. V. Flambaum, K. Beloy, and A. Derevianko, Phys. Rev. A 82, 062513 (2010).
  * (70) S. Biswas, A. Das, A. Bhowmik, and S. Majumder, Mon. Not. R. Astron. Soc. 477, 5605 (2018).
  * (71) A. Kramida, Y. Ralchenko, J. Reader, and NIST ASD Team (2019). NIST Atomic Spectra Database (ver. 5.7.1), [Online]. Available: https://physics.nist.gov/asd [2020, October 26].
  * (72) W. R. Johnson, Z. W. Liu, and J. Sapirstein, At. Data Nucl. Data Tables 64, 279 (1996).
  * (73) U. I. Safronova, Phys. Rev. A 82, 022504 (2010).
  * (74) E. Arimondo, M. Inguscio, and P. Violino, Rev. Mod. Phys. 49, 31 (1977).
  * (75) T. Kuga, Y. Torii, N. Shiokawa, and T. Hirano, Phys. Rev. Lett. 78, 4713 (1997).
  * (76) S. A. Kennedy, G. W. Biedermann, J. T. Farrar, T. G. Akin, S. P. Krzyzewski, and E. R. I. Abraham, Optics Communications 321, 110 (2014).
  * (77) J. Arlt, K. Dholakia, J. Soneson and E. M. Wright Physical Review A 63, 063602 (2001).
  * (78) E. M. Wright, J. Arlt, and K. Dholakia, Phys. Rev. A 63, 013608 (2000).


School of Geographical Science and Urban Planning, Arizona State University,
Tempe, AZ, USA yuanyuantian@asu.eduhttps://orcid.org/0000-0002-5137-1296School
of Geographical Science and Urban Planning, Arizona State University, Tempe,
AZ, USA<EMAIL_ADDRESS>Yuanyuan and Wenwen National Science Foundation (NSF)
under award OIA-2033521.

###### Acknowledgements.

Thank Mark Schildhauer and Rui Zhu at University of California Santa Barbara,
and Cogan Shimizu at Kansas State University for ontology engineering and
comments. 1st International Workshop on Methods, Models, and Resources for
Geospatial Knowledge Graphs and GeoAI GeoKG&GeoAI GeoKG & GeoAI 2021 September
27, 2021 Poznań, Poland 06

# GeoAI for Knowledge Graph Construction: Identifying Causality Between
Cascading Events to Support Environmental Resilience Research

Yuanyuan Tian  Wenwen Li111corresponding author

###### Abstract

Knowledge graph technology is considered a powerful and semantically enabled
solution to link entities, allowing users to derive new knowledge by reasoning
data according to various types of reasoning rules. However, in building such
a knowledge graph, events modeling, such as that of disasters, is often
limited to single, isolated events. The linkages among cascading events are
often missing in existing knowledge graphs. This paper introduces our GeoAI
(Geospatial Artificial Intelligence) solutions to identify causality among
events, in particular, disaster events, based on a set of spatially and
temporally-enabled semantic rules. Through a use case of causal disaster
events modeling, we demonstrated how these defined rules, including theme-
based identification of correlated events, spatiotemporal co-occurrence
constraint, and text mining of event metadata, enable the automatic extraction
of causal relationships between different events. Our solution enriches the
event knowledge base and allows for the exploration of linked cascading events
in large knowledge graphs, therefore empowering knowledge query and discovery.

###### keywords:

Knowledge Graph, Cascading Events, Disaster, Causality

###### category:

##  1 Introduction

Global warming and climate change exacerbate the frequency and severity of
global disasters. Since 1980, nearly 300 weather and climate disaster events
have hit the U.S., each causing over 1 billion dollars of economic loss, and
the National Ocean and Atmospheric Agency (NOAA) released a recent Billion-
Dollar Disaster Report showing 2020 as a historic year of extreme climate
disasters [8]. There is an urgent need to improve our understanding of both
historical and incoming disastrous events to develop disaster preparation and
response solutions to increase our urban and ecosystem’s resilience to these
events [5].

In recent years, knowledge graphs (KG) have become a popular approach for
building machine-understandable semantic models of entities and their
interrelationships. These entities could be a static object, such as a
building or an event, such as a hurricane. Building a large knowledge graph
with linked data can enable a new paradigm for question answering and
knowledge discovery [4]. In disaster modeling, knowledge graphs also play a
key role in integrating multi-modal data for improved disaster management and
response [2]. Since disasters often have a cause-and-effect relationship that
reflects the interconnections between events, linking cascading events has
become a central task in the semantic and formal modeling of disasters.
However, existing efforts have been mostly centered on modeling single events,
and the interrelations among them which are substantially missing [7]. The
entity alignment technique has the potential to discover identical or related
geospatial entities and harmonize heterogeneous datasets[9], yet barely
deliver the interlinking value of cascading disaster events.

This paper aims to address the limitation in knowledge graph modeling of
cascading events, by developing a GeoAI [3] solution with a set of semantic
rules for causality identification. These rules build upon each other and help
to derive from (1) thematically correlated events (i.e., those clustered in a
single large event collection) to (2) spatially and temporally co-occurred
events (i.e., cascading events that often occur in sequence and are spatially
correlated), to (3) semantically filtered events by mining the events metadata
to further determine their causal relationship. We use disaster modeling as
our use case to demonstrate the effectiveness of the proposed approach.

##  2 Methodology

We use severe storms and extreme events data to build a knowledge graph (KG)
that allows for the exploration of cascading severe weather events. The goal
is to create linkages in a disaster domain KG to allow semantic query and
knowledge inference. Three semantic rules are defined to identify causality
among events and answer pilot questions, such as “What are cascading E1 and E2
severe events during a specific disaster D? ” In this query, E1 and E2 are
words related to event types, D is the name of a named disaster. Rule #1
identifies thematically correlated events, rule #2 finds disaster events that
match with report information according to spatial and temporal co-occurrence
constraints, and rule #3 uses metadata text mining to determine the causal
relationships between events.

###  2.1 Data and Graph Building

The Storm Event Database covers the U.S. from January 1950 to April 2021 and
contains event reports collected by National Weather Service (NWS). This
database is used as our test dataset, in which an episode is an entire storm
system and includes various types of events. An event represents a
meteorological event leading to fatalities, injuries, and other kinds of
damages [6]. Multiple episodes can be related to a disaster such as Hurricane
Sandy, and each episode can encompass multiple severe events. There are 48
different types of meteorological events, from localized thunderstorms,
tornadoes, and flash floods to regional events such as hurricanes and winter
storms. Each record documents event attributes related to an event type, event
location, event start and end time, weather conditions, and event and episode
narratives. Impacts also include crop and property damage as well as direct
and indirect death and injury. A disaster ontology is constructed using the
original data structure and Sensor-Observation-Sampling-Actuator ontology to
achieve an explicit hierarchy. Then, the storm event records are populated
into the ontology to build the KG.

###  2.2 Thematically Correlated Event Identification

To deconstruct the pilot question, the first step is to understand the E1 and
E2 in the question. That human-readable question should be translated to a
general term rather than the explicit event type named E1 such as “flood”
because there are other types such as “flash flood” which also satisfy the
question’s semantics. Even domain experts may not be familiar with this
dataset’s event type classification standard. Thus, we introduce the concept
of theme to interpret E1 and E2 as “E1-themed” and “E2-themed" correlated
event types, respectively. For example, “hurricane-themed” and “flood-themed”
events are defined as: “Flood-themed” event types - Flood, FlashFlood,
CoastalFlood, LakeshoreFlood; “Hurricane-themed” event types - Hurricane,
Heavy Rain, High Surf, Marine Hurricane Typhoon, Marine Tropical Depression,
Marine Tropical Storm, Sneakerwave, StormSurgeTide, Tropical Depression,
Tropical Storm. Through this query expansion method, all thematically relevant
events can be retrieved from the dataset.

###  2.3 Disaster Matching According to Spatial and Temporal Co-occurrence
Constraint

The next rule is to support the identification of severe events related to a
disaster named D. This is challenging due to the data structure. NWS assigned
a unique numerical ID to an episode or an event; however, the disaster name
was not specified as an attribute. We proposed a solution to match disasters
with events based on two constraints. First, we used a keyword to match the
disaster name with the episode or event narrative to identify the data records
containing the needed disaster information. However, matching by disaster name
is insufficient because some narratives include comparison description
sentences referring to a previous disaster. Thus, we added the second
constraint based on spatial and temporal co-occurrences of the known events.
The event geometry or occurring location should overlap with the regions that
have reported impacts during a given time period. The event reports had
timestamps with a duration measured to the minute, indicating that the events
had short-term impacts. Since some impacts can occur before the event’s formed
date or after the dissipated date, a seven-day time buffer is added to the
duration coverage.

###  2.4 Metadata Text Mining

Metadata text mining is used to find causality among "E1–themed" and
"E2–themed" event narratives to empower linkage discovery and prediction. The
first task is to build a knowledge base containing terms that represent
causality and extract connected events. A thesaurus is built by integrating
causal keywords and phrases such as "lead to", "result in”, and "induce"
collected from Merriam-Webster Dictionary, Lexico, and [1] as connector
candidates. The next task is to construct event causality networks from event
narratives. Specific casual events are extracted and linked, then a
hierarchical causality generation method [10] is used to uncover the high-
level causality patterns on top of the specific one.

Figure 1: A pair of linked events: event 1, a flood and event 2, heavy rain.

##  3 Result

According to the semantically defined rules, the event network that has causal
relationships can be found. For example, “What are cascading flood and
hurricane events during Hurricane Sandy?” Figure 1 depicts findings of a pair
of occurrences of events from Hurricane Sandy’s devastation. A beige node is a
flooding event and the red node refers to a heavy rain event. Each resultant
event is depicted by time, location, and damage caused. The flood narrative
shows the correct name for Sandy and claims that the heavy rains caused minor
flooding.

##  4 Conclusions and Future Work

This paper develops a new GeoAI based semantic analysis approach that connects
the cascading disaster events. The method is capable of discovering the
interconnections between many types of severe weather events, their effects on
the surrounding environment, and impacts on human society. Leveraging the
information about interlinked events, researchers and decision makers could
gain a better understanding of compound hazards and cascading events for
disaster preparation and response. This will increase urban and environmental
resilience to incoming disasters. In the future, we will optimize the spatial
and temporal co-occurrence constraint by testing different topological
measures and temporal buffers. Also, refine the methodology to include deep
learning based link prediction for finding the causal relationships to
establish the semantically enriched linked data in an automatic manner. We
will also extend the methodological framework to support the construction of a
large-scale knowledge graph with multiple types of disasters, and add
quantitative evaluation results.

## References

  * [1] Amir AghaKouchak, Laurie S Huning, Felicia Chiang, Mojtaba Sadegh, Farshid Vahedifard, Omid Mazdiyasni, Hamed Moftakhari, and Iman Mallakpour.  How do natural hazards cascade to cause disasters?, 2018.
  * [2] Chao Fan, Cheng Zhang, Alex Yahja, and Ali Mostafavi.  Disaster city digital twin: A vision for integrating artificial and human intelligence for disaster management.  International Journal of Information Management, 56:102049, 2021\.
  * [3] Wenwen Li.  GeoAI: Where machine learning and big data converge in GIScience.  Journal of Spatial Information Science, (20):71–77, 2020.
  * [4] Wenwen Li, Miaomiao Song, and Yuanyuan Tian.  An ontology-driven cyberinfrastructure for intelligent spatiotemporal question answering and open knowledge discovery.  ISPRS International Journal of Geo-Information, 8(11):496, 2019\.
  * [5] Dayton Marchese, Erin Reynolds, Matthew E Bates, Heather Morgan, Susan Spierre Clark, and Igor Linkov.  Resilience and sustainability: Similarities and differences in environmental management applications.  Science of the total environment, 613:1275–1283, 2018.
  * [6] JD Murphy.  Storm data preparation (national weather service instruction 10-1605).  Retrieved on May, 24, 2016.
  * [7] Gianluca Pescaroli and David Alexander.  A definition of cascading disasters and cascading effects: Going beyond the “toppling dominos” metaphor.  Planet@ risk, 3(1):58–67, 2015.
  * [8] Adam B Smith.  Us billion-dollar weather and climate disasters, 1980-present (ncei accession 0209268).  NOAA National Centers for Environmental Information. https://doi. org/10.25921/STKW-7W73, 2020.
  * [9] Kai Sun, Yingjie Hu, Jia Song, and Yunqiang Zhu.  Aligning geographic entities from historical maps for building knowledge graphs.  International Journal of Geographical Information Science, pages 1–30, 2020.
  * [10] Sendong Zhao, Quan Wang, Sean Massung, Bing Qin, Ting Liu, Bin Wang, and ChengXiang Zhai.  Constructing and embedding abstract event causality networks from text snippets.  In Proceedings of the Tenth ACM International Conference on Web Search and Data Mining, pages 335–344, 2017.