Magnetic Sublevel Independent Magic and Tune-Out Wavelengths of the Alkaline-Earth Ions

Abstract

Light shift in a state due to the applied laser in an atomic system vanishes at tune-out wavelengths (${\lambda }_T$s). Similarly, differential light shift in a transition vanishes at the magic wavelengths (${\lambda }_{magic}$s). In many of the earlier studies, values of the electric dipole (E1) matrix elements were inferred precisely by combining measurements and calculations of ${\lambda }_{magic}$. Similarly, the ${\lambda }_T$ values of an atomic state can be used to infer the E1 matrix element, as it involves dynamic electric dipole ($\alpha$) values of only one state whereas the ${\lambda }_{magic}$ values require evaluation of $\alpha$ values for two states. However, both the ${\lambda }_{magic}$ and ${\lambda }_T$ values depend on angular momenta and their magnetic components (M) of states. Here, we report the ${\lambda }_{magic}$ and ${\lambda }_T$ values of many $S_{1/2}$ and $D_{3/2,5/2}$ states, and transitions among these states of the Mg${}^{+}$, Ca${}^{+}$, Sr${}^{+}$ and Ba${}^{+}$ ions that are independent of M values. It is possible to infer a large number of E1 matrix elements of the above ions accurately by measuring these values and combining with our calculations.

1. Introduction

Singly charged alkaline-earth ions are the most eligible candidates for considering for high-precision measurements due to several advantages [1]. Except Be${}^{+}$ and Mg${}^{+}$, other alkaline-earth ions have two metastable states and most of the transitions among the ground and metastable states are accessible by lasers. This is why these ions are considered for carrying out high-precision measurements such as testing Lorentz symmetry violations [2,3,4], parity nonconservation effects [5], non-linear isotope shift effects [6], quantum information [7,8] and many more, including for the optical atomic clock experiments [9]. Above all, optical lattices blended with unique features of optical transitions in the ionic system lead to the revolution in the clock frequency states. Since the confinement of Mg${}^{+}$ ions in a monochromatic optical dipole trap has become feasible experimentally for several ms [10], the pathways to implement these ions to realize optical lattice clocks have been opened up due to the fact that ions provide more accurate atomic clocks since various systematics in the ions can be controlled easily [11,12].

One of the major systematics in the aforementioned high-precision measurements is the Stark shift due to the employed laser, which depends on the frequency of the laser. The solution to this problem was suggested by Katori et al. [13], who proposed that the trapping laser can be tuned to wavelengths at which differential ac Stark shifts of the transitions can vanish [13]. These wavelengths were coined as magic wavelengths (${\lambda }_{magic}$s) and are being popularly used in optical lattice clocks. The development of the technique to measure ${\lambda }_{magic}$s in the singly charged ${}^{40}$Ca${}^{+}$ ion [14] has opened up the possibilities to measure these wavelengths in the other ions, such as Al${}^{+}$, Hg${}^{+}$, Yb${}^{+}$, Cd${}^{+}$, In${}^{+}$ and so on, that are being considered in many high-precision measurements. Further, the infrared magic wavelength has been identified recently for all optical trapping of ${}^{40}$Ca${}^{+}$ ion clock [15]. The stability of optical lattices can be enhanced by optical cavities in various dimensions while availing the coordinated choice of ${\lambda }_{magic}$s [16] and state-dependent potentials [17], which can further be implemented in the all-optical confinement of these ions. In quantum state engineering [18], ${\lambda }_{magic}$s provide an opportunity to extract accurate values of oscillator strengths [19], which are particularly important for correct stellar modeling and analysis of spectral lines identified in the spectra of stars and other heavenly bodies so as to infer fundamental stellar parameters [20,21].

Apart from the magic trapping condition, where light shift in two internal states is identical, another well-known limiting case is where light shift in one state vanishes. This case is known as a tune-out condition [22]. Applications of such tune-out wavelengths (${\lambda }_T$) lie in novel cooling techniques of atoms [23], selective addressing and manipulation of quantum states [24,25,26], precision measurement of atomic structures [27,28,29,30,31,32] and precise estimation of oscillator strength ratios [33]. Additionally, tune-out conditions are powerful tools for the evaporative cooling of optical lattices [34] and, hence, are important for experimental explorations.

In one of the experiments pertaining to magic wavelengths of alkaline-earth ions, Liu et al. demonstrated the existence of magic wavelengths for a single trapped ${}^{40}$Ca${}^{+}$ ion [14], whereas Jiang et al. evaluated magic wavelengths of Ca${}^{+}$ ions for linearly and circularly polarized light using the relativistic configuration interaction plus core polarization (RCICP) approach [35,36]. Recently, Chanu et al. proposed a model to trap Ba${}^{+}$ ions by inducing an ac Stark shift using a 653 nm linearly polarized laser [37]. Kaur et al. reported magic wavelengths for $n{S}_{1/2}-n{P}_{1/2,3/2}$ and $n{S}_{1/2}-m{D}_{3/2,5/2}$ transitions in alkaline-earth-metal ions using linearly polarized light [16], whereas Jiang et al. located magic and tune-out wavelengths for Ba${}^{+}$ ions using the RCICP approach [38]. Despite having a large number of applications, these magic wavelengths suffer a setback because of their dependency on the magnetic-sublevels (M) of the atomic systems. Linearly polarized light has been widely used for the trapping of atoms and ions as it is free from the contribution of the vector component in the interaction between atomic states and electric fields. However, the magic wavelengths thus identified are again magnetic-sublevel dependent for the transitions involving states with angular momenta greater than $1/2$. On the other hand, the cooling of ions using circularly polarized light requires magnetic-sublevel selections. In order to circumvent this M dependency of magic wavelengths, a magnetic-sublevel-independent strategy for the trapping of atoms and ions was proposed by Sukhjit et al. [39]. Later on, Kaur et al. implemented a similar technique to compute magic and tune-out wavelengths independent of magnetic sublevels M for different $n{S}_{1/2}-(n-1)D_{3/2,5,2}$ transitions in Ca${}^{+}$, Sr${}^{+}$ and Ba${}^{+}$ ions corresponding to n = 4 for Ca${}^{+}$, 5 for Sr${}^{+}$ and 6 for Ba${}^{+}$ ion [40].

In addition to the applications of ${\lambda }_{magic}$ in getting rid of differential Stark shift in a transition, they are also being used to infer the electric dipole (E1) matrix elements of many allowed transitions in different atomic systems [14,38,41,42]. The basic procedure of these studies is that the ${\lambda }_{magic}$ values are calculated by fine-tuning of the magnitudes of dominantly contributing E1 matrix elements to reproduce their measured values. Then, the set of the E1 matrix elements that gives rise to the best matched ${\lambda }_{magic}$ values are considered as the recommended E1 matrix elements. However, calculations of these ${\lambda }_{magic}$ values of a transition demand the determination of the dynamic E1 polarizabilities ($\alpha$) of both the states. In view of this, use of the ${\lambda }_T$ values of a given atomic state can be advantageous, as they involve dynamic $\alpha$ values of only one state. Furthermore, both the ${\lambda }_{magic}$ and ${\lambda }_T$ values depend on angular momenta and their magnetic components (M) of atomic states. This requires evaluation of scalar, vector and tensor components of the $\alpha$ values for states with angular momenta greater than $1/2$, which is very cumbersome. To circumvent this problem, we present here M-sublevel-independent ${\lambda }_{magic}$ and ${\lambda }_T$ values of many states and transitions involving a number of $S_{1/2}$ and $D_{3/2,5/2}$ states in the alkaline-earth metal ions from Mg${}^{+}$ through Ba${}^{+}$ that can be inferred to the E1 matrix elements more precisely. The basic idea behind this work is to use the calculated E1 matrix elements to predict the M-independent ${\lambda }_{magic}$ and ${\lambda }_T$ values of the above ions. Theoretical determinations of these ${\lambda }_{magic}$ and ${\lambda }_T$ are very sensitive to the magnitudes of a few principal E1 matrix elements. When measurements of these quantities are available, by combining them with our calculations it is possible to fine tune the E1 matrix element values to higher precision. At this stage, our reported ${\lambda }_{magic}$ and ${\lambda }_T$ values will be useful to guide the experimentalists to carry out these measurements. The measurements can be achieved by setting up experiments suitably by fixing the polarization and quantization angles of the applied lasers. To validate our results for the transitions involving high-lying states, we compared the values of our ${\lambda }_{magic}$ and ${\lambda }_T$ values for the ground to the metastable states of the considered alkaline-earth ions with the previously reported values.

The paper is organized as follows: In Section 2, we provide underlying theory and Section 3 describes the method of evaluation of the calculated quantities. Section 4 discusses the obtained results, while concluding the study in Section 5. Unless we have stated explicitly, physical quantities are given in atomic units (a.u.).

2. Theory

The electric field $\mathcal{E}$(r,$t)$ associated with a general plane electromagnetic wave can be represented in terms of complex polarization vector $\widehat{\chi }$ and the real wave vector k by the following expression [43]

$$\mathcal{E}(\mathbf{r},t)=\frac{1}{2}\mathcal{E}\widehat{\chi }{e}^{-\iota (\omega t-\mathbf{k}.\mathbf{r})}+c.c.,$$

(1)

where $c.c.$ is the complex conjugate of the preceding term. Assuming $\widehat{\chi }$ to be real and adopting the coordinate system as presented in Figure 1, the polarization vector can be expressed as [39]

$$\widehat{\chi }=e^{\iota \sigma }(cos\varphi {\widehat{\chi }}_{maj}+\iota sin\varphi {\widehat{\chi }}_{min}),$$

(2)

where ${\widehat{\chi }}_{maj}$ and ${\widehat{\chi }}_{min}$ denote the real components of the polarization vector $\widehat{\chi }$, $\sigma$ is the real quantity denoting the arbitrary phase and $\varphi$ is analogous to degree of polarization A such that $A=sin\left(2\varphi \right)$. For linearly polarized light, $\varphi =0$ whereas $\varphi$ takes the value either $\pi /4$ or $3\pi /4$ for circularly polarized light, which further defines $A=0$ for linearly polarized and $A=1(-1)$ for right-hand (left-hand) circularly polarized light [43]. As shown in Figure 1, this coordinate system follows

$$co{s}^2{\theta }_p=co{s}^2\varphi co{s}^2{\theta }_{maj}+si{n}^2\varphi si{n}^2{\theta }_{min}$$

(3)

and

$${\theta }_{maj}+{\theta }_{min}=\frac{\pi }{2}.$$

(4)

Here, ${\theta }_p$ is the angle between quantization axis ${\widehat{\chi }}_B$ and direction of polarization vector $\widehat{\chi }$ and the parameters ${\theta }_{maj}$ and ${\theta }_{min}$ are the angles between respective unit vectors and $\widehat{{\chi }_B}$.

When an atomic system is subjected to the above electric field and the magnitude of $\mathcal{E}$ is small, a shift in the energy of its $n^{th}$ level (Stark shift) can be given by

$$\begin{array}{c}\hfill \delta E_n^K\simeq -\frac{1}{2}{\alpha }_n^K\left(\omega \right){\left|\mathcal{E}\right|}^2,\end{array}$$

(5)

where ${\alpha }_n^K\left(\omega \right)$ is known as the second-order electric dipole (E1) polarizability and the superscript K denotes angular momentum of the state, which can be atomic angular momentum J or hyperfine-level angular momentum F. Depending upon polarization, dynamic dipole polarizability ${\alpha }_n^K\left(\omega \right)$ can be expressed as

$$\begin{array}{c}{\alpha }_n^K\left(\omega \right)={\alpha }_{nS}^K\left(\omega \right)+\beta \left(\chi \right)\frac{M_K}{2K}{\alpha }_{nV}^K\left(\omega \right)\\ +\gamma \left(\chi \right)\frac{3M_K^2-K(K+1)}{K(2K-1)}{\alpha }_{nT}^K\left(\omega \right),\end{array}$$

(6)

where ${\alpha }_{nS}^K$, ${\alpha }_{nV}^K$ and ${\alpha }_{nT}^K$ are the scalar, vector and tensor components of the polarizability, respectively. The expression can be defined on the basis of the coordinate system provided in the Figure 1. Geometrically, values for $\beta \left(\chi \right)$ and $\gamma \left(\chi \right)$ in their elliptical form are given as [39,43]

$$\beta \left(\chi \right)=\iota (\widehat{\chi }\times {\widehat{\chi }}^{*}).{\widehat{\chi }}_B=Acos{\theta }_k$$

(7)

and

$$\gamma \left(\chi \right)=\frac{1}{2}\left[3({\widehat{\chi }}^{*}.{\widehat{\chi }}_B)(\widehat{\chi }.{\widehat{\chi }}_B)-1\right]=\frac{1}{2}\left(3co{s}^2{\theta }_p-1\right),$$

(8)

where ${\theta }_k$ is the angle between direction of propagation $\mathbf{k}$ and ${\widehat{\chi }}_B$. Substitution of $\beta \left(\chi \right)$ and $\gamma \left(\chi \right)$ from Equations (7) and (8) reforms the expression for dipole polarizability to

$$\begin{array}{c}{\alpha }_n^K\left(\omega \right)={\alpha }_{nS}^K\left(\omega \right)+Acos{\theta }_k\frac{M_K}{2K}{\alpha }_{nV}^K\left(\omega \right)\\ +\left(\frac{3co{s}^2{\theta }_p-1}{2}\right)\frac{3M_K^2-K(K+1)}{K(2K-1)}{\alpha }_{nT}^K\left(\omega \right)\end{array}$$

(9)

with the azimuthal quantum number $M_K$ of the respective angular momentum K.

Thus, it is obvious from Equation (6) that the ${\alpha }_n^K$ values of two states have to be same if we intend to find ${\lambda }_{magic}$ for the transition involving both the states. Since the above expression for ${\alpha }_n^K$ has $M_K$ dependency, the ${\lambda }_{magic}$ become $M_K$ dependent. In order to remove $M_K$ dependency, one can choose $M_K=0$ sublevels but in the atomic states of the alkaline-earth ions they are non-zero while isotopes with integer nuclear spin of the alkaline-earth ions $M_K$s are again non-zero. To address this, a suitable combination of the $\beta \left(\chi \right)$ and $\gamma \left(\chi \right)$ parameters needs to be chosen such that $cos{\theta }_k=0$ and $co{s}^2{\theta }_p=\frac{1}{3}$, which are feasible to achieve in an experiment by setting ${\theta }_k,{\widehat{\chi }}_{maj}$ and $\varphi$ values, as demonstrated in Ref. [39]. In such a scenario, the ${\lambda }_{magic}$ values can depend on the scalar part only by suppressing the vector and tensor components of ${\alpha }_n^K$; i.e., the net differential Stark effect of a transition occurring from the J to $J^{\prime }$ states will be given by

$$\delta E_{J{J}^{\prime }}=-\frac{1}{2}\left[{\alpha }_{nS}^J\left(\omega \right)-{\alpha }_{nS}^{J^{\prime }}\left(\omega \right)\right]{\mathcal{E}}^2.$$

(10)

This has an additional advantage that the differential Stark effects at an arbitrary electric field become independent of the choice of atomic or hyperfine levels in a given atomic system, as the scalar component of ${\alpha }_n^J$ and ${\alpha }_n^F$ are the same. Again, the same choice of ${\lambda }_{magic}$ values will be applicable to both the atomic and hyperfine levels in a high-precision experiment.

3. Method of Evaluation

Determination of ${\alpha }_n^J$ values requires accurate calculations of E1 matrix elements. For the computation of E1 matrix elements, we need accurate atomic wave functions of the alkaline-earth ions. We employed here a relativistic all-order method to determine the atomic wave functions of the considered atomic systems, whose atomic states have a closed-core configuration with an unpaired electron in the valence orbital. Detailed descriptions of our all-order method can be found in Refs. [44,45,46,47]; however, a brief outline of the same is also provided here for the completeness.

Our all-order method follows the relativistic coupled-cluster (RCC) theory ansätz

$$|{\psi }_v\rangle =e^S|{\varphi }_v\rangle ,$$

(11)

where $|{\varphi }_v\rangle$ represents the mean-field wave function of the state v and constructed as [48]

$$|{\varphi }_v\rangle =a_v^{†}|0_c\rangle ,$$

(12)

where $|0_c\rangle$ represents the Dirac–Hartree–Fock (DHF) wave function of the closed core. Subscript v represents the valence orbital of the considered state. In our calculations, we consider only linear terms in the singles and doubles approximation of the RCC theory (SD method) by expressing [48]

$$|{\psi }_v\rangle =(1+S_1+S_2+...)|{\varphi }_v\rangle ,$$

(13)

where $S_1$ and $S_2$ depict terms corresponding to the single and double excitations, respectively, that can further be written in terms of second quantization creation and annihilation operators as follows [49]

$$\begin{array}{c}\hfill S_1=\sum _{ma}{\rho }_{ma}{a}_m^{†}{a}_a+\sum _{m\ne v}{\rho }_{mv}{a}_m^{†}{a}_v\end{array}$$

(14)

and

$$\begin{array}{c}\hfill S_2=\frac{1}{2}\sum _{mnab}{\rho }_{mnab}{a}_m^{†}{a}_n^{†}{a}_b{a}_a+\sum _{mna}{\rho }_{mnva}{a}_m^{†}{a}_n^{†}{a}_a{a}_v,\end{array}$$

(15)

where indices m and n range over all possible virtual orbitals, and indices a and b range over all occupied core orbitals. The coefficients ${\rho }_{ma}$ and ${\rho }_{mv}$ represent excitation coefficients of the respective single excitations for the core and the valence electrons, respectively, whereas ${\rho }_{mnab}$ and ${\rho }_{mnva}$ depict double excitation coefficients for the core and the valence electrons, respectively. These amplitudes are calculated in an iterative procedure [50], due to which they include electron correlation effects to all-order.

Hence, atomic wave function of the considered states in the alkaline-earth ions are expressed as [48,51]:

$$\begin{array}{cc}|{\psi }_v{\rangle }_{SD}& \hfill =\left[1+\sum _{ma}{\rho }_{ma}{a}_m^{†}{a}_a+\frac{1}{2}\sum _{mnab}{\rho }_{mnab}{a}_m^{†}{a}_n^{†}{a}_b{a}_a\right.\\ & \hfill \left.+\sum _{m\ne v}{\rho }_{mv}{a}_m^{†}{a}_v+\sum _{mna}{\rho }_{mnva}{a}_m^{†}{a}_n^{†}{a}_a{a}_v\right]|{\varphi }_v\rangle .\end{array}$$

(16)

To improve the calculations further and understand the importance of contributions from the triple excitations in the RCC theory, we take into account important core and valence triple excitations through the perturbative approach over the SD method (SDpT method) by redefining the wave function expression as [48]

$$\begin{array}{cc}|{\psi }_v{\rangle }_{SDpT}& \hfill =|{\psi }_v{\rangle }_{SD}+\left[\frac{1}{6}\sum _{mnrab}{\rho }_{mnrvab}{a}_m^{†}{a}_n^{†}{a}_r^{†}{a}_b{a}_a{a}_v\right.\\ & \hfill \left.+\frac{1}{18}\sum _{mnrabc}{\rho }_{mnrabc}{a}_m^{†}{a}_n^{†}{a}_r^{†}{a}_c{a}_b{a}_a\right]|{\varphi }_v\rangle .\end{array}$$

(17)

After obtaining the wave functions of the interested atomic states, we evaluate the E1 matrix elements between states $|{\psi }_v\rangle$ and $|{\psi }_w\rangle$ as [49]

$$\begin{array}{c}\hfill D_{wv}=\frac{\langle {\psi }_w\left|D\right|{\psi }_v\rangle }{\sqrt{\langle {\psi }_w|{\psi }_w\rangle \langle {\psi }_v|{\psi }_v\rangle }},\end{array}$$

(18)

where $D=-e{\Sigma }_j{\mathbf{r}}_j$ is the E1 operator with ${\mathbf{r}}_j$ being the position of jth electron [51]. The resulting expression of numerator of Equation (18) includes the sum of the DHF matrix elements $z_{wv}$; twenty correlation terms of the SD method that are linear or quadratic functions of excitation coefficients ${\rho }_{mv}$, ${\rho }_{ma}$, ${\rho }_{mnva}$ and ${\rho }_{mnab}$; and their core counterparts [45].

In the sum-over-states approach, expression for the scalar dipole polarizability is given by

$${\alpha }_v\left(\omega \right)=\frac{2}{3(2J_v+1)}\sum _{v\ne w}\frac{(E_v-E_w)|\langle {\psi }_v\left|\right|D\left|\right|{\psi }_w{\rangle |}^2}{{(E_v-E_w)}^2-{\omega }^2},$$

(19)

where $\langle {\psi }_v\left|\right|D\left|\right|{\psi }_w\rangle$ is the reduced matrix element for the transition occurring between the states involving the valence orbitals v and w. Here, we dropped the superscript J in the dipole polarizability notation for the brevity. For convenience, we divide the entire contribution to ${\alpha }_n\left(\omega \right)$ for any state v, into three parts, as

$${\alpha }_v={\alpha }_{v,c}+{\alpha }_{v,vc}+{\alpha }_{v,val}$$

(20)

where $c,vc$ and $val$ corresponds to core, valence-core and valence contributions arising due to the correlations among the core orbitals, core-valence orbitals and valence-virtual orbitals, respectively [52]. Due to much smaller magnitudes, the core and core-valence contributions are calculated using the DHF method. The dominant contributions will arise valence from ${\alpha }_{v,val}$ due to small energy denominators. Again, the high-lying states will not contribute to ${\alpha }_{v,val}$ owing to large energy denominators. Thus, we calculate E1 matrix elements only among the low-lying excited states and refer the contributions as ‘Main’. Contributions from the less contributing high-lying states are referred as ‘Tail’ and are estimated again using the DHF method. To reduce the uncertainties in the estimations of Main contributions, we used experimental energies of the states from the National Institute of Science and Technology atomic database (NIST AD) [53].

4. Results and Discussion

The precise computation of magic and tune-out wavelengths requires the accurate determination of E1 matrix elements as well as dipole polarizabilities. In our work, we used E1 matrix elements available on Portal for High-Precision Atomic Data and Computation [54] and NIST Atomic Spectra Database [53], respectively.

We listed resonance transitions, magic wavelengths and their corresponding polarizabilities along with their uncertainties for magnetic-sublevel independent $nS-mD$ transitions for alkaline-earth ions from Mg${}^{+}$ through Ba${}^{+}$ along with their comparison with the available literature in the

Table 1. Magic wavelengths ${\lambda }_{magic}$ (in nm) with their corresponding polarizability ${\alpha }_n\left(\omega \right)$ (in a.u.) for $(3,4)S_{1/2}-3D_{3/2,5/2}$ and $4S_{1/2}-4D_{3/2}$ transitions in Mg${}^{+}$ ion.

$3{\mathit{S}}_{1/2}-3{\mathit{D}}_{3/2}$				$3{\mathit{S}}_{1/2}-3{\mathit{D}}_{5/2}$
Resonance	${\mathbf{\lambda }}_{\mathit{res}}$	${\mathbf{\lambda }}_{\mathit{magic}}$	${\mathbf{\alpha }}_{\mathit{magic}}$	Resonance	${\mathbf{\lambda }}_{\mathit{res}}$	${\mathbf{\lambda }}_{\mathit{magic}}$	${\mathbf{\alpha }}_{\mathit{magic}}$
$3D_{3/2}\to 6P_{3/2}$	$292.92$			$3D_{5/2}\to 5F_{5/2}$	$310.56$
		$313.89\left(2\right)$	$168.72\left(13\right)$			$313.86\left(2\right)$	$168.85\left(13\right)$
$3D_{3/2}\to 5P_{1/2}$	$385.15$			$3D_{5/2}\to 5P_{3/2}$	$384.920$
		$385.300\left(1\right)$	$73.51\left(7\right)$			$385.101\left(1\right)$	$73.59\left(6\right)$
		$757.8\left(1.5\right)$	$40.43\left(5\right)$
$3D_{3/2}\to 5P_{3/2}$	$1091.83$			$3D_{5/2}\to 4F_{5/2}$	$448.24$
		$1092.436\left(1\right)$	$37.41\left(5\right)$			$756.7\left(1.5\right)$	$40.45\left(5\right)$
$3D_{3/2}\to 4P_{1/2}$	$1095.48$			$3D_{5/2}\to 4P_{3/2}$	$1091.72$
$4S_{1/2}-3D_{3/2}$				$4S_{1/2}-3D_{5/2}$
$4S_{1/2}\to 5P_{3/2}$	$361.48$			$4S_{1/2}\to 5F_{7/2}$	$310.56$
		$361.26\left(1\right)$	$-202.1\left(1.3\right)$			$344.87\left(47\right)$	$-160.00\left(13\right)$
$4S_{1/2}\to 5P_{1/2}$	$361.66$			$4S_{1/2}\to 5P_{3/2}$	$361.48$
		$361.627\left(1\right)$	$-203.2\left(1.3\right)$			$361.626\left(1\right)$	$-203.8\left(1.3\right)$
$4S_{1/2}\to 4P_{3/2}$	$922.08$			$4S_{1/2}\to 5P_{1/2}$	$361.66$
		$923.812\left(1\right)$	$-140.74\left(97\right)$
$4S_{1/2}\to 4P_{1/2}$	$924.68$			$3D_{5/2}\to 5P_{3/2}$	$384.93$
						$385.408\left(4\right)$	$-163.67\left(13\right)$
$3D_{3/2}\to 4P_{3/2}$	$1091.83$			$3D_{5/2}\to 4F_{5/2}$	$448.24$
		$1092.377\left(1\right)$	$1976.9\left(1.5\right)$
$3D_{3/2}\to 4P_{1/2}$	$1095.48$			$4S_{1/2}\to 4P_{3/2}$	$922.08$
		$1132.53\left(6\right)$	$1681.4\left(1.3\right)$			$923.812\left(1\right)$	$-144.72\left(98\right)$
				$4S_{1/2}\to 4P_{1/2}$	$924.68$
				$3D_{5/2}\to 4P_{3/2}$	$1091.72$
						$1128.42\left(6\right)$	$1706.3\left(1.3\right)$
$4S_{1/2}-4D_{3/2}$				$4S_{1/2}-4D_{5/2}$
$4D_{3/2}\to 7F_{5/2}$	$526.58$			$4D_{5/2}\to 7F_{5/2,7/2}$	$526.57$
		$591.48\left(1\right)$	$-422.01\left(31\right)$			$591.34\left(1\right)$	$-421.69\left(31\right)$
$4D_{3/2}\to 7P_{3/2}$	$591.83$			$4D_{5/2}\to 7P_{3/2}$	$591.81$
		$591.8603\left(3\right)$	$-422.86\left(31\right)$
$4D_{3/2}\to 7P_{1/2}$	$591.98$
		$616.11\left(22\right)$	$-482.44\left(35\right)$			$616.02\left(23\right)$	$-482.19\left(35\right)$
$4D_{3/2}\to 6F_{5/2}$	$634.87$			$4D_{5/2}\to 6F_{7/2}$	$634.85$
$4P_{1/2}\to 4D_{3/2}$	$787.92$
		$789.499\left(4\right)$	$-1578.9\left(1.2\right)$
$4P_{3/2}\to 4D_{3/2}$	$789.82$			$4P_{3/2}\to 4D_{5/2}$	$789.85$
$4D_{3/2}\to 6P_{3/2}$	$811.78$			$4D_{5/2}\to 6P_{3/2}$	$811.75$
		$811.8186\left(1\right)$	$-1973.6\left(1.5\right)$			$812.118\left(1\right)$	$-1980.0\left(1.5\right)$
						$844.63\left(10\right)$	$-2964.8\left(2.2\right)$
$4D_{3/2}\to 6P_{1/2}$	$812.83$
		$812.648\left(1\right)$	$-1991.3\left(1.5\right)$
		$843.61\left(11\right)$	$-2921.4\left(2.2\right)$
$4S_{1/2}\to 4P_{3/2}$	$922.08$			$4S_{1/2}\to 4P_{3/2}$	$922.08$
		$923.839\left(1\right)$	$-4957.9\left(6.5\right)$			$923.839\left(1\right)$	$-4975.8\left(6.4\right)$
$4S_{1/2}\to 4P_{1/2}$	$924.68$			$4S_{1/2}\to 4P_{1/2}$	$924.68$
$4D_{3/2}\to 5F_{5/2}$	$963.51$			$4D_{5/2}\to 5F_{7/2}$	$963.45$
		$1006.10\left(13\right)$	$3585.5\left(2.7\right)$			$1005.86\left(13\right)$	$3594.8\left(2.7\right)$

through Table 4, respectively. Results reported by Kaur et al. [40] were obtained by similar analyses, but contributions from some of the high-lying states were neglected in that work. Furthermore, we used excitation energies for most of the states listed in the NIST AD compared to Kaur et al. This resulted in more precise values of dipole polarizabilities, hence further improving the magic and tune-out wavelengths.

Further discussion regarding the magic wavelengths is provided in the Section 4.1 for the considered alkaline-earth ions. Furthermore, we discussed our results for tune-out wavelengths in the Section 4.2 along with the comparison of our results with respect to the available theoretical data.

4.1. Magic Wavelengths

4.1.1. Mg${}^{+}$

In Table 1, we tabulated our results for magic wavelengths and their corresponding dipole polarizabilities for $(3,4)S_{1/2}-3D_{3/2,5/2}$ and $4S_{1/2}-4D_{3/2}$ transitions. Figure 2a demonstrates scalar dipole polarizabilities of $3S_{1/2}$ and $3D_{3/2,5/2}$ states of Mg${}^{+}$ ion with respect to the wavelength of the external field. It can be perceived from the figure that a number of magic wavelengths at the crossings of the scalar polarizabilities’ curves of the corresponding state have been predicted for the transition. As can be seen from Table 1, a total of 4 magic wavelengths were found for $3S-3D_{3/2}$ transition, whereas the $3S-3D_{5/2}$ transition shows a total of 3 magic wavelengths in the range 300–1250 nm, out of which no magic wavelength was found to exist in visible spectrum. However, all the magic wavelengths enlisted in Table 1 support red-detuned trap.

Figure 2b represents the plot of scalar dipole polarizabilities of $4S$ and $4D_{3/2,5/2}$ states against the wavelength of the external field. It can also be assessed from Table 1 that there exists a total of nine magic wavelengths in the considered wavelength range for $4S-4D_{3/2}$ transition, whereas only five magic wavelengths are seen for the $4S-4D_{5/2}$ transition. However, in both cases, all the magic wavelengths except those around 616 nm, 844 nm and 1006 nm are close to resonance, thereby making them unsuitable for further use. However, out of these three values, ${\lambda }_{magic}$ at 616 nm lies in the visible region and is far-detuned with considerable deep potential. Hence, we recommend this magic wavelength for the trapping of Mg${}^{+}$ ions for both $4S$–$4D_{3/2,5/2}$ transitions for further experimentations in optical clock applications.

Figure 2c demonstrates the magic wavelengths for the M${}_J$-independent scheme for $4S$–$3D_{3/2,5/2}$ transitions for Mg${}^{+}$ ions along with their corresponding scalar dynamic polarizabilities. According to Table 1, it can be realized that none of the magic wavelengths for these transitions lie within the visible spectrum of electromagnetic radiations. However, all of these magic wavelengths support a red-detuned trap, except $1132.53\left(6\right)$ nm and $1128.42\left(6\right)$ nm for $4S$-$3D_{3/2}$ and $4S-3D_{5/2}$ transitions, respectively, support far blue-detuned traps and are found to be useful for experimental demonstrations.

4.1.2. Ca${}^{+}$

We considered $4S-3D_{3/2,5/2}$ and $5S-(4,3)D_{3/2,5/2}$ transitions for locating the magic wavelengths in Ca${}^{+}$ ion. We tabulated magic wavelengths for these transitions along with the comparison of ${\lambda }_{magic}$s with the only available results for $4S-3D_{3/2,5/2}$ in

Table 2. Magic wavelengths ${\lambda }_{magic}$ (in nm) with their corresponding polarizability ${\alpha }_n\left(\omega \right)$ (in a.u.) for $(4,5)S_{1/2}-3D_{3/2,5/2}$ and $5S_{1/2}-4D_{3/2,5/2}$ transitions in Ca${}^{+}$ ion and their comparison with the available literature.

$4{\mathit{S}}_{1/2}-3{\mathit{D}}_{3/2}$				$4{\mathit{S}}_{1/2}-3{\mathit{D}}_{5/2}$
Resonance	${\mathbf{\lambda }}_{\mathit{res}}$	${\mathbf{\lambda }}_{\mathit{magic}}$	${\mathbf{\alpha }}_{\mathit{magic}}$	Resonance	${\mathbf{\lambda }}_{\mathit{res}}$	${\mathbf{\lambda }}_{\mathit{magic}}$	${\mathbf{\alpha }}_{\mathit{magic}}$
$4S_{1/2}\to 4P_{3/2}$	$393.48$			$4S_{1/2}\to 4P_{3/2}$	$393.47$
		$395.795\left(3\right)$	$5.6\left(1.6\right)$			$395.795\left(3\right)$	$5.6\left(1.6\right)$
		$395.82\left(3\right)$40]	$4.90$ [40]			$395.82\left(2\right)$ [40]	$95.87$ [40]
$4S_{1/2}\to 4P_{1/2}$	$396.96$			$4S_{1/2}\to 4P_{1/2}$	$396.96$
$3D_{3/2}\to 4P_{3/2}$	$850.04$			$3D_{5/2}\to 4P_{3/2}$	$854.44$
		$852.42\left(2\right)$	$95.67\left(26\right)$			$1011.90\left(90\right)$	$88.89\left(25\right)$
		$852.45\left(2\right)$ [40]	$4.20$ [40]			$1014.10\left(3\right)$ [40]	$89.01$ [40]
$3D_{3/2}\to 4P_{1/2}$	$866.45$
		$1029.0\left(1.2\right)$	$88.39\left(25\right)$
		$1029.7\left(2\right)$ [40]	$88.55$ [40]
$5S_{1/2}-3D_{3/2}$				$5S_{1/2}-3D_{5/2}$
$4P_{1/2}\to 5S_{1/2}$	$370.71$			$4P_{1/2}\to 5S_{1/2}$	$370.71$
		$371.76\left(1\right)$	$6.7\left(1.6\right)$			$371.76\left(1\right)$	$6.7\left(1.6\right)$
$4P_{3/2}\to 5S_{1/2}$	$373.80$			$4P_{3/2}\to 5S_{1/2}$	$373.80$
$5S_{1/2}\to 6P_{3/2}$	$447.33$			$5S_{1/2}\to 6P_{3/2}$	$447.33$
		$447.385\left(5\right)$	$3.0\left(1.6\right)$			$447.385\left(5\right)$	$2.9\left(1.6\right)$
$5S_{1/2}\to 6P_{1/2}$	$448.07$			$5S_{1/2}\to 6P_{1/2}$	$448.07$
		$448.088\left(2\right)$	$2.9\left(1.6\right)$			$448.088\left(2\right)$	$2.9\left(1.6\right)$
		$847.69\left(2\right)$	$-1089.4\left(2.8\right)$			$845.78\left(4\right)$	$-1079.8\left(2.8\right)$
$3D_{3/2}\to 4P_{3/2}$	$850.04$			$3D_{5/2}\to 4P_{3/2}$	$854.44$
		$860.22\left(7\right)$	$-1154.7\left(3.0\right)$
$3D_{3/2}\to 4P_{1/2}$	$866.45$
$5S_{1/2}\to 5P_{3/2}$	$1184.22$			$5S_{1/2}\to 5P_{3/2}$	$1184.22$
		$1191.59\left(1\right)$	$58.9\left(1.6\right)$		$1191.59\left(1\right)$		$56.8\left(1.5\right)$
$5S_{1/2}\to 5P_{1/2}$	$1195.30$			$5S_{1/2}\to 5P_{1/2}$	$1195.30$
$5S_{1/2}-4D_{3/2}$				$5S_{1/2}-4D_{5/2}$
$5S_{1/2}\to 6P_{3/2}$	$447.33$			$5S_{1/2}\to 6P_{3/2}$	$447.34$
		$447.52\left(2\right)$	$-204.2\left(2.4\right)$			$447.52\left(2\right)$	$-205.3\left(2.4\right)$
$5S_{1/2}\to 6P_{1/2}$	$448.07$			$5S_{1/2}\to 6P_{1/2}$	$448.07$
		$448.16\left(2\right)$	$-204.7\left(2.4\right)$			$448.16\left(1\right)$	$-205.8\left(2.4\right)$
		$471.22\left(24\right)$	$-279.71\left(85\right)$			$471.67\left(23\right)$	$-279.85\left(85\right)$
$4D_{3/2}\to 5F_{5/2}$	$471.81$			$4D_{5/2}\to 5F_{5/2,7/2}$	$472.23$
		$565.51\left(2\right)$	$-355.55\left(91\right)$			$566.03\left(13\right)$	$-356.17\left(92\right)$
$4D_{3/2}\to 6P_{3/2}$	$565.53$			$4D_{5/2}\to 6P_{3/2}$	$566.15$
		$566.67\left(17\right)$	$-356.93\left(92\right)$
$4D_{3/2}\to 6P_{1/2}$	$566.71$
$4D_{3/2}\to 4F_{5/2}$	$891.45$			$4D_{5/2}\to 4F_{7/2}$	$892.98$
$5S_{1/2}\to 5P_{3/2}$	$1184.22$			$5S_{1/2}\to 5P_{3/2}$	$1184.22$
		$1191.56\left(1\right)$	$776.1\left(8.1\right)$			$1191.56\left(1\right)$	$779.9\left(7.9\right)$
$5S_{1/2}\to 5P_{1/2}$	$1195.30$			$5S_{1/2}\to 5P_{1/2}$	$1195.302$

. In addition, we plotted scalar dipole polarizabilities against wavelengths for these transitions in Figure 2d–f, correspondingly. According to Table 2, it is determined that, subsequently, three and two magic wavelengths exist between 393 nm and 1030 nm for $4S-3D_{3/2,5/2}$ transitions. In both cases, except $1029.0\left(1.2\right)$ nm and $1011.90\left(90\right)$ nm magic wavelengths, which are far-detuned, all other magic wavelengths are close to resonances and are not suitable for laser trapping.

During analysis, six and five magic wavelengths were located for the $5S-(3,4)D_{3/2}$ and $5S-(3,4)D_{5/2}$ transitions, respectively. It was also determined that all the magic wavelengths are approximately same for both $5S-4D_{3/2}$ and $5S-4D_{5/2}$ transitions. Moreover, ${\lambda }_{magic}$s around 845 nm, 847 nm and 860 nm share deep trapping potential for blue-detuned traps and, hence, are further recommended for configuring feasible traps. ${\lambda }_{magic}$ at $1191.56$ nm, identified in the infrared region for both $5S-4D_{3/2,5/2}$ transitions, is the only magic wavelength that supports the red-detuned trap. In addition, the polarizability for this wavelength is sufficient enough for creating an ion trap at reasonable laser power. To validate our results, we also compared our results with the results provided only for $4S-3D_{3/2,5/2}$ in Ref. [40], and noticed that the results for these transitions are in good agreement with only less than $1\%$ variation w.r.t. obtained results.

4.1.3. Sr${}^{+}$

Figure 3a–c demonstrate M${}_J$-independent dynamic dipole polarizability versus wavelength plots for $(6,5)S_{1/2}-4D_{3/2,5/2}$ and $6S_{1/2}-5D_{3/2,5/2}$ transitions for Sr${}^{+}$ ion. The results corresponding to these figures are listed in

Table 3. Magic wavelengths ${\lambda }_{magic}$ (in nm) with their corresponding polarizability ${\alpha }_n\left(\omega \right)$ (in a.u.) along with their comparison with the available literature for $(5,6)S_{1/2}-4D_{3/2,5/2}$ and $6S_{1/2}-5D_{3/2,5/2}$ transitions in Sr${}^{+}$ ion.

$5{\mathit{S}}_{1/2}-4{\mathit{D}}_{3/2}$				$5{\mathit{S}}_{1/2}-4{\mathit{D}}_{5/2}$
Resonance	${\mathbf{\lambda }}_{\mathit{res}}$	${\mathbf{\lambda }}_{\mathit{magic}}$	${\mathbf{\alpha }}_{\mathit{magic}}$	Resonance	${\mathbf{\lambda }}_{\mathit{res}}$	${\mathbf{\lambda }}_{\mathit{magic}}$	${\mathbf{\alpha }}_{\mathit{magic}}$
$5S_{1/2}\to 5P_{3/2}$	$407.89$			$5S_{1/2}\to 5P_{3/2}$	$407.89$
		$417.00\left(5\right)$	$15.3\left(2.1\right)$			$417.00\left(4\right)$	$15.2\left(2.1\right)$
		$416.9\left(3\right)$ [40]	$14.47$ [40]			$416.9\left(3\right)$ [40]	$13.3$ [40]
$5S_{1/2}\to 5P_{1/2}$	$421.67$			$5S_{1/2}\to 5P_{1/2}$	$421.67$
$4D_{3/2}\to 5P_{3/2}$	$1003.94$			$4D_{5/2}\to 5P_{3/2}$	$1003.01$
		$1014.68\left(26\right)$	$108.70\left(90\right)$
		$1014.6\left(2\right)$ [40]	$108.35$ [40]
$4D_{3/2}\to 5P_{1/2}$	$1091.79$
$6S_{1/2}-4D_{3/2}$				$6S_{1/2}-4D_{5/2}$
$5P_{1/2}\to 6S_{1/2}$	$416.27$			$5P_{1/2}\to 6S_{1/2}$	$416.30$
		$421.47\left(5\right)$	$15.0\left(2.1\right)$			$421.47\left(5\right)$	$14.9\left(2.1\right)$
$5P_{3/2}\to 6S_{1/2}$	$430.67$			$5P_{3/2}\to 6S_{1/2}$	$430.67$
$6S_{1/2}\to 7P_{3/2}$	$474.37$			$6S_{1/2}\to 7P_{3/2}$	$474.37$
		$474.61\left(1\right)$	$11.3\left(2.1\right)$			$474.61\left(1\right)$	$10.9\left(2.1\right)$
$6S_{1/2}\to 7P_{1/2}$	$477.49$			$6S_{1/2}\to 7P_{1/2}$	$477.49$
		$477.549\left(0\right)$	$11.1\left(2.1\right)$			$477.557\left(5\right)$	$10.7\left(2.1\right)$
		$1002.40\left(3\right)$	$-2470.0\left(7.5\right)$			$1025.19\left(17\right)$	$-2858.0\left(8.8\right)$
$4D_{3/2}\to 5P_{3/2}$	$1003.94$			$4D_{5/2}\to 5P_{3/2}$	$1033.01$
		$1087.35\left(9\right)$	$-4653\left(15\right)$
$4D_{3/2}\to 5P_{1/2}$	$1091.79$			$6S_{1/2}\to 6P_{3/2}$	$1201.73$
$6S_{1/2}\to 6P_{3/2}$	$1201.73$					$1230.05\left(6\right)$	$170.2\left(3.6\right)$
		$1230.02\left(6\right)$	$223.4\left(4.2\right)$
$6S_{1/2}\to 6P_{1/2}$	$1244.84$			$6S_{1/2}\to 6P_{1/2}$	$1244.84$
$6S_{1/2}-5D_{3/2}$				$6S_{1/2}-5D_{5/2}$
$5D_{3/2}\to 5F_{5/2}$	$562.45$			$5D_{5/2}\to 5F_{5/2}$	$565.20$
		$643.871\left(2\right)$	$-528.6\left(1.5\right)$			$647.41\left(2\right)$	$-534.6\left(1.6\right)$
$5D_{3/2}\to 7P_{3/2}$	$643.88$			$5D_{5/2}\to 7P_{3/2}$	$647.49$
		$649.49\left(2\right)$	$-538.2\left(1.6\right)$
$5D_{3/2}\to 7P_{1/2}$	$649.65$
$6S_{1/2}\to 6P_{3/2}$	$1201.73$			$6S_{1/2}\to 6P_{3/2}$	$1201.73$
		$1233.61\left(5\right)$	$-6756\left(29\right)$			$1233.06\left(5\right)$	$-5620\left(23\right)$
$6S_{1/2}\to 6P_{1/2}$	$1244.84$			$6S_{1/2}\to 6P_{1/2}$	$1244.84$
$5D_{3/2}\to 4F_{5/2}$	$1297.85$			$5D_{5/2}\to 4F_{5/2}$	$1312.62$
		$1411.9\left(1.5\right)$	$4382\left(13\right)$
				$5D_{5/2}\to 4F_{7/2}$	$1312.84$
						$1448.4\left(1.6\right)$	$3812\left(11\right)$

. Only two magic wavelengths were traced for the $5S-4D_{3/2}$ transition, whereas only one magic wavelength exists for the $5S-4D_{5/2}$ transition. According to Table 3, for the $6S-4D_{3/2}$ transition, three magic wavelengths exist below 480 nm, with a dynamic polarizability of value less than 15 a.u.; however, another three ${\lambda }_{magic}$s, lie between 1000 nm and 1231 nm. The ${\lambda }_{magic}$s at $1002.401$ nm and $1087.35$ nm support blue-detuned traps with sufficiently high polarizabilities for the experimental trapping of Sr${}^{+}$ ions. For the $6S-4D_{5/2}$ transition, five magic wavelengths were located between 420 nm and 1250 nm, out of which the magic wavelengths at $421.47\left(5\right)$ nm, $474.61\left(1\right)$ nm, $477.549\left(0\right)$ nm and $1230.05\left(6\right)$ nm follow red-detuned traps, whereas the only magic wavelength at $1025.19\left(17\right)$ nm, with corresponding $\alpha =-2858.0\left(8.8\right)$ a.u., supports a blue-detuned trap, which can be useful for experimental purposes. We recommend this magic wavelength of Sr${}^{+}$ ion for the $6S$-$4D_{5/2}$ transition. Moreover, it is also observed that all the magic wavelengths for these two transitions lie between the same resonance transitions and are closer to each other. Threfore, it is probable to trap Sr${}^{+}$ ion for both of these transitions with same magic wavelength.

Table 3 also shows that there are four magic wavelengths that lie within the wavelength range of 640 nm to 1450 nm for the $6S-5D_{3/2}$ transition. It is also observed that three out of four magic wavelengths for $6S-5D_{3/2}$ transition support blue-detuned traps; however, the ${\lambda }_{magic}=1233.61\left(5\right)$ nm at ${\alpha }_{magic}=-6756\left(24\right)$ a.u. is recommended for experimental purposes as it is far-detuned and a high value of dipole polarizability indicates deep trapping potential. On the other hand, only three magic wavelengths were identified for the $6S-5D_{5/2}$ transition in Sr${}^{+}$ ion with two supporting blue-detuned traps. Two out of these ${\lambda }_{magic}$s, i.e., $1233.06\left(5\right)$ nm and $1448.4\left(1.6\right)$ nm, are located at higher wavelength ranges, with deep potentials for their respective favourable blue- and red-detuned traps. Therefore, both of these values are recommended for further experimental studies. Moreover, we compared our magic wavelengths for $5S-4D_{3/2,5/2}$ transitions with respect to the available literature in the same table. It is seen that our reported values are in excellent approximation with the results obtained by Kaur et al. [40], with a variation by less than $0.05\%$. Unfortunately, we could not find any data related to other transitions to carry out the comparison with. Hence, it can be concluded from the comparison of available data that our results are promising and can be used for further prospective calculations of atomic structures and atomic properties of this ion.

4.1.4. Ba${}^{+}$

The results for magic wavelengths for $6S-5D_{3/2,5/2}$, $7S-5D_{3/2,5/2}$ and $7S-6D_{3/2,5/2}$ transitions in Ba${}^{+}$ ion are tabulated in

Table 4. Magic wavelengths ${\lambda }_{magic}$ (in nm) with their corresponding polarizability ${\alpha }_n\left(\omega \right)$ (in a.u.) along with their comparison with the available literature for $(6,7)S_{1/2}-5D_{3/2,5/2}$ and $7S_{1/2}-6D_{3/2,5/2}$ transitions in Ba${}^{+}$ ions.

$6{\mathit{S}}_{1/2}-5{\mathit{D}}_{3/2}$				$6{\mathit{S}}_{1/2}-5{\mathit{D}}_{5/2}$
Resonance	${\mathbf{\lambda }}_{\mathit{res}}$	${\mathbf{\lambda }}_{\mathit{magic}}$	${\mathbf{\alpha }}_{\mathit{magic}}$	Resonance	${\mathbf{\lambda }}_{\mathit{res}}$	${\mathbf{\lambda }}_{\mathit{magic}}$	${\mathbf{\alpha }}_{\mathit{magic}}$
$6S_{1/2}\to 6P_{3/2}$	$455.53$			$6S_{1/2}\to 6P_{3/2}$	$455.53$
		$480.71\left(2\right)$	$-4.1\left(1.6\right)$			$480.76\left(2\right)$	$-8.3\left(1.4\right)$
		$480.6\left(5\right)$ [40]	$-2.89$ [40]
$6S_{1/2}\to 6P_{1/2}$	$493.55$			$6S_{1/2}\to 6P_{1/2}$	$493.55$
$5D_{3/2}\to 6P_{3/2}$	$585.53$			$5D_{3/2}\to 6P_{3/2}$	$614.34$
		$588.32\left(1\right)$	$330.15\left(60\right)$			$653.17\left(\right)35$	$247.90\left(57\right)$
		$588.4\left(3\right)$ [40]	$329.33$ [40]			$695.7\left(3\right)$ [40]	$219.4$ [40]
$5D_{3/2}\to 6P_{1/2}$	$649.87$
		$693.46\left(48\right)$	$221.91\left(56\right)$
		$655.50\left(3\right)$ [40]	$244.89$ [40]
$7S_{1/2}-5D_{3/2}$				$7S_{1/2}-5D_{5/2}$
$6P_{1/2}\to 7S_{1/2}$	$452.62$			$6P_{1/2}\to 7S_{3/2}$	$452.62$
		$466.95\left(13\right)$	$0.5\left(1.6\right)$			$466.88\left(9\right)$	$-2.6\left(1.5\right)$
$6P_{3/2}\to 7S_{1/2}$	$490.13$			$6P_{3/2}\to 7S_{1/2}$	$490.13$
$7S_{1/2}\to 8P_{3/2}$	$518.49$			$7S_{1/2}\to 8P_{3/2}$	$518.49$
		$518.79\left(2\right)$	$-22.9\left(1.5\right)$			$518.79\left(2\right)$	$-31.9\left(1.5\right)$
$7S_{1/2}\to 8P_{1/2}$	$526.75$			$7S_{1/2}\to 8P_{1/2}$	$526.75$
		$526.779\left(6\right)$	$-28.7\left(1.5\right)$			$601.37\left(6\right)$	$-548.8\left(1.6\right)$
		$583.76\left(1\right)$	$-548.6\left(2.4\right)$
$5D_{3/2}\to 6P_{3/2}$	$585.53$			$5D_{5/2}\to 6P_{3/2}$	$614.34$
		$638.75\left(5\right)$	$-573.9\left(2.4\right)$
$5D_{3/2}\to 6P_{1/2}$	$649.87$
$7S_{1/2}\to 7P_{3/2}$	$1306.14$			$7S_{1/2}\to 7P_{3/2}$	$1306.14$
		$1380.83\left(24\right)$	$59.8\left(1.4\right)$			$1380.83\left(24\right)$	$59.2\left(\right)1.3$
$7S_{1/2}\to 7P_{1/2}$	$1421.54$			$7S_{1/2}\to 7P_{1/2}$	$1421.54$
$7S_{1/2}-6D_{3/2}$				$7S_{1/2}-6D_{5/2}$
$7S_{1/2}\to 8P_{3/2}$	$518.49$			$7S_{1/2}\to 8P_{3/2}$	$518.49$
		$519.03\left(11\right)$	$-376\left(34\right)$			$519.07\left(10\right)$	$-405\left(31\right)$
$7S_{1/2}\to 8P_{1/2}$	$526.750$			$7S_{1/2}\to 8P_{1/2}$	$526.75$
		$526.84\left(12\right)$	$-483\left(43\right)$			$526.84\left(6\right)$	$-480\left(43\right)$
		$531.0\left(1.0\right)$	$-664.2\left(4.0\right)$			$532.8\left(2.4\right)$	$-655.2\left(3.8\right)$
$6D_{3/2}\to 6F_{5/2}$	$536.28$			$6D_{5/2}\to 6F_{7/2}$	$539.31$
						$542.17\left(7\right)$	$-613.1\left(3.2\right)$
				$6D_{5/2}\to 6F_{5/2}$	$542.26$
						$645.33\left(10\right)$	$-580.8\left(2.5\right)$
$6D_{3/2}\to 8P_{3/2}$	$637.25$			$6D_{5/2}\to 8P_{3/2}$	$645.70$
						$735.65\left(92\right)$	$-728.3\left(3.2\right)$
$6D_{3/2}\to 8P_{1/2}$	$649.77$			$6D_{5/2}\to 5F_{7/2}$	$871.32$
		$744.0\left(2.2\right)$	$-746.4\left(3.2\right)$			$889.20\left(20\right)$	$-1211.6\left(5.5\right)$
$6D_{3/2}\to 5F_{5/2}$	$874.02$			$6D_{5/2}\to 5F_{5/2}$	$889.99$
$7S_{1/2}\to 7P_{3/2}$	$1306.14$			$7S_{1/2}\to 7P_{3/2}$	$1306.14$
		$1381.25\left(35\right)$	$-78\left(77\right)$			$1381.39\left(33\right)$	$-125\left(77\right)$
$7S_{1/2}\to 7P_{1/2}$	$1421.54$			$7S_{1/2}\to 7P_{1/2}$	$1421.54$

. As per Figure 3d and Table 4, a maximum number of magic wavelengths were located between 480 and 700 nm. It is also observed that the magic wavelengths that lie between $6S-6P_{1/2}$ and $6S-6P_{3/2}$ resonant transitions support blue-detuned trap; however, the dynamic dipole polarizability corresponding to these magic wavelengths are too small to trap Ba${}^{+}$ ions at these wavelengths. A total of six magic wavelengths were found for the $7S-5D_{3/2}$ transition, out of which two lie in the vicinity of 526 nm. The sharp intersection of the polarizability curves of the involved states of transition lie at $583.76\left(1\right)$ nm, $638.75\left(5\right)$ nm and $1380.83\left(24\right)$ nm. Similarly, four magic wavelengths were identified for the $7S-5D_{5/2}$ transition; however, unlike the $7S-5D_{3/2}$ transition, no magic wavelength was identified in the vicinity of 600 to 1300 nm. It was also determined that three out of these four ${\lambda }_{magic}$s, support a blue-detuned trap, although the trapping potentials for these traps are not deep enough for further consideration in experimentations.

Table 4 also compiles the magic wavelengths for $7S-5D_{3/2,5/2}$ transitions and shows that there exists six and four magic wavelengths for $7S-5D_{3/2}$ and $7S-5D_{5/2}$ transition, respectively. It is also seen that the magic wavelengths between $6P_{3/2}-7S$ and $7S-8P_{3/2}$ as well as the $5D_{3/2}-6P_{1/2}$ and $7S-7P_{3/2}$ transitions seem to be missing, as shown in Figure 3e. It is also observed that the magic wavelength at $466.95\left(13\right)$ nm and $1380.83\left(24\right)$ nm are slightly red-shifted; nevertheless, the ${\lambda }_{magic}$ at $638.75\left(5\right)$ nm in the visible region supports a blue-detuned trap, and can have sufficient trap depth at reasonable laser power.

Similarly, the magic wavelengths and their corresponding dynamic dipole polarizability along with their comparison with available literature is also provided in the same table for $7S-6D_{3/2,5/2}$ transitions. The same was demonstrated graphically in Figure 3f, which includes a total of thirteen magic wavelengths in all for the considered transitions. It is also examined that no magic wavelength exists between $6D_{3/2}-6F_{5/2}$ and $6D_{3/2}-8P_{1/2}$ resonances. Unlike the $7S-6D_{3/2}$ transition, around eight magic wavelengths were located between $7S-8P_{3/2}$ and $7S-7P_{1/2}$ resonances, and all of them support blue-detuned traps. Moreover, magic wavelengths at $532.8\left(2.4\right)$ nm, $735.65\left(92\right)$ nm and $1381.39\left(33\right)$ nm are expected to be more promising for experiments due to sufficient trap depths for reasonable-power lasers. However, on the comparison of our results for $6S-5D_{3/2,5/2}$ transitions for Ba${}^{+}$ ion, we observed that all the magic wavelengths agree well with the results obtained by Kaur et al., in Ref. [40], except the last magic wavelengths that were identified at 693 nm and 653 nm for $6S-5D_{3/2}$ and $6S-5D_{5/2}$ transitions.

4.2. Tune-Out Wavelengths

We illustrated tune-out wavelengths for different states of the considered transitions in the alkaline-earth ions along with their comparison with the already available literature in

Table 5. Tune-out wavelengths ${\lambda }_T$ (in nm) of various states of Mg${}^{+}$, Ca${}^{+}$, Sr${}^{+}$ and Ba${}^{+}$ ions and their comparison with available literature.

Mg${}^{+}$			Ca${}^{+}$			Sr${}^{+}$			Ba${}^{+}$
State	${\mathbf{\lambda }}_{\mathit{T}}$	Others	State	${\mathbf{\lambda }}_{\mathit{T}}$	Others	State	${\mathbf{\lambda }}_{\mathit{T}}$	Others	State	${\mathbf{\lambda }}_{\mathit{T}}$	Others
$3S_{1/2}$	$102.6058\left(6\right)$		$4S_{1/2}$	$165.04\left(3\right)$		$5S_{1/2}$	$417.04\left(5\right)$	$417.04\left(6\right)$ [55]	$6S_{1/2}$	$200.035\left(1\right)$
								$417.025$ [40]
	$102.696\left(8\right)$			$165.26\left(4\right)$
	$124.071\left(7\right)$			$395.796\left(3\right)$	$395.80\left(2\right)$ [55]					$202.51\left(9\right)$
					$395.796$ [40]
	$280.1136\left(2\right)$	$280.110\left(9\right)$ [55]								$480.66\left(2\right)$	$480.63\left(24\right)$ [55]
											$480.66\left(18\right)$ [38]
											$480.596$ [40]
$3D_{3/2}$	$317.082\left(59\right)$		$3D_{3/2}$	$212.930\left(5\right)$		$4D_{3/2}$	$185.498\left(2\right)$		$5D_{3/2}$	$224.675\left(6\right)$
	$384.9615\left(4\right)$			$213.25\left(2\right)$			$192.95\left(95\right)$			$468.61\left(61\right)$	$472.461$ [40]
	$385.3401\left(9\right)$			$494.37\left(63\right)$	$492.752$ [40]		$242.622\left(7\right)$			$597.93\left(15\right)$	$597.983$ [40]
	$812.03\left(97\right)$			$852.75\left(3\right)$	$852.776$ [40]		$606.47\left(53\right)$	$598.633$ [40]
	$1092.436\left(9\right)$						$1018.91\left(38\right)$	$1018.873$ [40]
$3D_{5/2}$	$317.003\left(6\right)$		$3D_{5/2}$	$170.63\left(63\right)$		$4D_{5/2}$	$193.78\left(78\right)$		$5D_{5/2}$	$193.32\left(16\right)$
	$385.143\left(9\right)$			$213.16\left(3\right)$			$242.56\left(1\right)$			$198.43\left(57\right)$
	$810.69\left(31\right)$			$493.13\left(87\right)$	$482.642$ [40]		$593.0\left(2.0\right)$	$585.677$ [40]		$225.63\left(1\right)$
										$234.773\left(3\right)$
										$459.57\left(57\right)$	$509.687$ [40]
$4S_{1/2}$	$279.1332\left(4\right)$		$5S_{1/2}$	$287.179\left(7\right)$		$6S_{1/2}$	$334.71\left(71\right)$		$7S_{1/2}$	$347.63\left(5\right)$
	$293.2134\left(2\right)$			$299.64\left(64\right)$			$362.58\left(2\right)$			$363.36\left(64\right)$
	$361.524\left(2\right)$			$309.13\left(5\right)$			$378.373\left(3\right)$			$394.63\left(6\right)$
	$361.691\left(1\right)$			$309.303\left(9\right)$			$421.40\left(5\right)$			$466.94\left(13\right)$
				$371.754\left(9\right)$			$474.62\left(1\right)$			$518.78\left(2\right)$
				$447.386\left(5\right)$			$477.559\left(5\right)$			$526.778\left(6\right)$
				$448.088\left(2\right)$			$1230.15\left(6\right)$			$1381.01\left(23\right)$
				$1191.59\left(1\right)$
$4D_{3/2}$	$554.11\left(12\right)$		$4D_{3/2}$	$317.850\left(5\right)$		$5D_{3/2}$	$407.4377\left(3\right)$		$6D_{3/2}$	$544.38\left(62\right)$
	$592.70\left(3\right)$			$334.65\left(1\right)$			$434.69\left(23\right)$			$1316.93\left(93\right)$
	$1331.53\left(53\right)$			$371.2822\left(5\right)$			$481.52\left(2\right)$
				$375.78\left(3\right)$			$573.80\left(56\right)$
				$471.96\left(6\right)$			$649.82\left(2\right)$
$4D_{5/2}$	$507.25\left(1\right)$		$4D_{5/2}$	$376.05\left(3\right)$		$5D_{5/2}$	$436.25\left(22\right)$		$6D_{5/2}$	$547.53\left(53\right)$
	$553.95\left(95\right)$			$472.38\left(6\right)$			$455.827\left(4\right)$			$889.04\left(23\right)$
	$592.55\left(3\right)$			$566.15\left(10\right)$			$577.18\left(57\right)$			$1287.17\left(83\right)$
	$811.9701\left(4\right)$						$647.60\left(2\right)$
	$1329.45\left(55\right)$						$1312.6247\left(1\right)$

. To locate these M${}_J$-independent tune-out wavelengths, we evaluated the scalar dipole dynamic polarizabilities of these states for considered alkaline-earth ions and identified those values of $\lambda$ for which polarizability vanished. It is also highlighted that, in Mg${}^{+}$ ion, all the tune-out wavelengths identified for $3S_{1/2}$ and $4S_{1/2}$ states lie in the UV region, whereas, for $(3,4)D_{3/2,5/2}$ states, a few tune-out wavelengths are located in visible range. Moreover, the largest ${\lambda }_T$ is identified for the $4D_{3/2}$ state at $1331.53\left(53\right)$ nm. Furthermore, only one tune-out wavelength, i.e., ${\lambda }_T=280.1136\left(2\right)$ nm for $3S_{1/2}$, could be compared with the result presented by Kaur et al. in Ref. [55] and it is seen that our result is in good accord with this value.

Similarly, we pointed out tune-out wavelengths for $n{S}_{1/2}$ and $(n-1)D_{3/2}$, $n=(4,5),(5,6)$ and $(6,7)$ states for Ca${}^{+}$, Sr${}^{+}$ and Ba${}^{+}$ ions, by identifying $\lambda$s at which their corresponding $\alpha$s tend to zero. Hence, it was perceived that, out of 25 tune-out wavelengths for all states of Ca${}^{+}$ ion, only seven of them lie within visible spectrum and, on comparison of different tune-out wavelengths for the $4S_{1/2}$ and $3D_{3/2}$ states of Ca${}^{+}$ ion, it was determined that all of these results are supported by the results obtained in Refs. [40,55]. However, one of the tune-out wavelength located at $493.13\left(87\right)$ nm for the $3D_{5/2}$ state of Ca${}^{+}$ ion seems have $2\%$ variation from the wavelength obtained by Kaur et al. in Ref. [40]. This results from the fact that the present study incorporates more precise E1 matrix elements for the high-lying transitions as well their excitation energies from the portal for High-Precision Atomic Data and Computation [54].

For Sr${}^{+}$ ion, the maximum number of tune-out wavelengths was identified out of all the considered alkaline-earth ions. It is also realized that most of these $\lambda$s lie within the visible spectrum of electromagnetic radiation, and are mostly comprised of all the ${\lambda }_T$ values corresponding to $5S_{1/2}$, $5D_{3/2}$ and $5D_{5/2}$ states. Additionally, during the comparison of these values with the results published in Refs. [40,55], it was determined that the tune-out wavelength at $417.04\left(6\right)$ nm for $5S_{1/2}$ as well as ${\lambda }_T=1018.91\left(38\right)$ nm for the $4D_{3/2}$ state agree well with the available results; howsoever, the tune-out wavelengths at $606.47\left(53\right)$ nm and $593.0\left(2.0\right)$ nm for $4D_{3/2}$ and $4D_{5/2}$ states, respectively, show a discrepancy of less than $2\%$, which lies within the error bar of $5\%$.

In the case of Ba${}^{+}$ ion, we located 23 tune-out wavelengths, which, in all, comprise of 10, 9 and 4 wavelengths in visible, UV and infrared regions, respectively. It is also highlighted that all the tune-out wavelengths that exist in visible region lie within the range 480 nm to 550 nm. We also compared our tune-out wavelengths for $6S_{1/2}$ and $5D_{3/2,5/2}$ states against available theoretical data in Refs. [38,40,55] and it was found that all the ${\lambda }_T$s except $468.61$ nm and $459.57$ nm, respectively, for $5D_{3/2}$ and $5D_{5/2}$ states show disparities of less than $1\%$, which lies within the considerable error limit.

5. Conclusions

We identified a number of reliable magnetic-sublevel-independent tune-out wavelengths of many $S_{1/2}$ and $D_{3/2,5/2}$ states, and magic wavelengths of different combinations of $S_{1/2}-D_{3/2,5/2}$ transitions in the alkaline-earth ions from Mg${}^{+}$ through Ba${}^{+}$. If they can be measured precisely, accurate values of many electric dipole matrix elements can be inferred by combining the experimental values of these quantities with our theoretical results. Most of the magic wavelengths found from this study show that they can be detected using the red- and blue-detuned traps. In fact, it is possible to perform many high-precision measurements by trapping the atoms at the reported tune-out and magic wavelengths of the considered transitions in the future, which can be applied to different metrological studies.