Magic wavelengths of Ca$^{+}$ ion for linearly and circularly polarized light

TL;DR

This paper calculates magic wavelengths for Ca$^{+}$ ion transitions under linearly and circularly polarized light using relativistic configuration interaction plus core polarization, providing data useful for precision measurements and quantum control.

Contribution

It presents new calculations of magic wavelengths for Ca$^{+}$ ion transitions, including for circular polarization, and suggests experimental measurements to determine oscillator strength ratios.

Findings

Magic wavelengths for linearly polarized light agree with previous results.

Additional magic wavelengths are identified for circularly polarized light.

Measurement near 850 nm can determine oscillator strength ratios.

Abstract

The dynamic dipole polarizabilities of the low-lying states of Ca$^{+}$ for linearly and circularly polarized light are calculated by using relativistic configuration interaction plus core polarization (RCICP) approach. The magic wavelengths, at which the two levels of the transitions have the same ac Stark shifts, for $4s$-$4p_{j,m}$ and $4s$-$3d_{j,m}$ magnetic sublevels transitions are determined. The present magic wavelengths for linearly polarized light agree with the available results excellently. The polarizability for the circularly polarized light has the scalar, vector and tensor components. The dynamic polarizability is different for each of magnetic sublevels of the atomic state. Additional magic wavelengths have been found for the circularly polarized light. We recommend that the measurement of the magic wavelength near 850 nm for $4s-4p_{\frac32,m=\pm\frac32,\pm\frac12}$ could be able to determine the oscillator strength ratio of $f_{4p_{\frac32} \to 3d_{\frac32}}$ and $f_{4p_{\frac32} \to 3d_{\frac52}}$.

Tables

Table 1. Table 1: Relativistic dipole polarizabilities (a.u.) of the ground and low-lying excited states of Ca + ion. The numbers in parentheses are uncertainties by introducing 0.5% uncertainties into the dominant matrix elements.

	$4s_{\frac{1}{2}}$	$3d_{\frac{3}{2}}$		$3d_{\frac{5}{2}}$		$4p_{\frac{1}{2}}$	$4p_{\frac{3}{2}}$
	${\alpha }_1$	${\alpha }_1$	${\alpha }_1^t$	${\alpha }_1$	${\alpha }_1^t$	${\alpha }_1$	${\alpha }_1$	${\alpha }_1^t$
Dipole
Present	75.46(72)	32.98(24)	$-$17.97(17)	32.80(24)	$-$25.28(24)	$-$2.98(11)	$-$1.12(10)	10.20(11)
DFCP Tang et al. (2013)	75.28	32.99	$-$17.88	32.81	$-$25.16	$-$2.774	$-$ 0.931	10.12
MBPT-SD Safronova and Safronova (2011a)	76.1(5)	32.0	$-$17.43(23)	32.0	$-$24.51(29)	$-$0.75	1.02(64)	10.31(28)
CICP Mitroy et al. (2008)	75.49	32.73	$-$17.64	32.73	$-$25.20	$-$2.032	$-$2.032	10.47
RCC Sahoo et al. (2009)	73.0(1.5)	28.5(1.5)	$-$15.87	29.5(1.0)	$-$22.49(5)
RCC-STO Sahoo et al. (2009)	74.3	31.6	$-$17.7	32.5	$-$25.5
f-sumsChang (1983)	75.3(4)

Table 2. Table 2: Comparison of static scalar polarizabilities (a.u.) for 4 ​ s 1 2 4 subscript 𝑠 1 2 4s_{\frac{1}{2}} and 3 ​ d 5 2 3 subscript 𝑑 5 2 3d_{\frac{5}{2}} states and blackbody radiation shift (Hz) for the 4 ​ s 1 2 − 3 ​ d 5 2 4 subscript 𝑠 1 2 3 subscript 𝑑 5 2 4s_{\frac{1}{2}}-3d_{\frac{5}{2}} transition of 40 Ca + ion at T=300 K.

	Present	Ref.Mitroy et al. (2008)	Ref.Sahoo et al. (2009)	Ref.Arora et al. (2007)	Ref.Champenois et al. (2005)	Ref.Kajita et al. (2005)	Expt.Guan et al. (2016)	Ref.Huang et al. (2012, 2011)
${\alpha }_0\left(4s_{\frac{1}{2}}\right)$	75.46(72)	75.49	73.0(1.5)	76.1(1.1)	76	73	76.1(1.1)
${\alpha }_0\left(3d_{\frac{5}{2}}\right)$	32.80(24)	32.73	29.5(1.0)	32.0(1.1)	31	23	31.8(3)
$\Delta \alpha$	42.66	42.76	43.5	44.1	45	40	44.3
$\eta \left(4s_{\frac{1}{2}}\right)$	$-3.0\left[-9\right]$
$\eta \left(3d_{\frac{5}{2}}\right)$	$-1.1\left[-8\right]$
BBR shift	0.367(39)	0.368	0.37(1)	0.38(1)	0.39(27)	0.4	0.35(0.009)111The temperature is 294.4 K in this experiment.	0.35

Table 3. Table 3: Magic wavelengths (in nm) for the transitions of low-lying states of Ca + for the linearly polarized light. The numbers in the parentheses are uncertainties calculated by assuming certain matrix elements have ± 2 plus-or-minus 2 \pm 2 % uncertainties.

Resonance	${\lambda }_{res}$	Present	Ref.Kaur et al. (2015)	Ref.Tang et al. (2013)	Exp.Liu et al. (2015)
Transitions	$4s_{\frac{1}{2}}-4p_{\frac{1}{2}}$
$4p_{\frac{1}{2}}-3d_{\frac{3}{2}}$	866.214
		691.24(12.29)	697.65	690.817(11.984)
$4p_{\frac{1}{2}}-4s_{\frac{1}{2}}$	396.847
		395.1788(377)	395.18	395.1807(14)
$4p_{\frac{1}{2}}-5s_{\frac{1}{2}}$	370.603
		368.0221(1412)	368.10	368.0149(901)
$4p_{\frac{1}{2}}-4d_{\frac{3}{2}}$	315.887
$4s_{\frac{1}{2}}-4p_{\frac{3}{2}},m=\frac{3}{2}$
$4p_{\frac{3}{2}}-3d_{\frac{5}{2}}$	854.209
		850.9217(15)	850.12
$4p_{\frac{3}{2}}-3d_{\frac{3}{2}}$	849.802
		672.89(15.33)	678.35	672.508(11.3150)
$4p_{\frac{1}{2}}-4s_{\frac{1}{2}}$	396.847
		395.7729(19)	395.77	395.774(10)
$4p_{\frac{3}{2}}-4s_{\frac{1}{2}}$	393.366
$4s_{\frac{1}{2}}-4p_{\frac{3}{2}},m=\frac{1}{2}$
$4p_{\frac{3}{2}}-3d_{\frac{5}{2}}$	854.209
		850.1164(1)	850.12
$4p_{\frac{3}{2}}-3d_{\frac{3}{2}}$	849.802
		687.51(10.33)	693.76	687.022(12.285)
$4p_{\frac{1}{2}}-4s_{\frac{1}{2}}$	396.847
		396.2297(218)	396.23	396.2315(13)
$4p_{\frac{3}{2}}-4s_{\frac{1}{2}}$	393.366
$4p_{\frac{3}{2}}-5s_{\frac{1}{2}}$	373.690
		369.6523(1849)	369.72	369.6393(1534)
$4p_{\frac{3}{2}}-4d_{\frac{3}{2}}$	318.128
$4s_{\frac{1}{2}}-3d_{\frac{3}{2}},m=\frac{3}{2}$
		887.28(3.52)	884.54	887.382(3.196)
$3d_{\frac{3}{2}}-4p_{\frac{1}{2}}$	866.214
$4s_{\frac{1}{2}}-4p_{\frac{1}{2}}$	396.847
		395.7951 (1)	395.79	395.7970(1)
$4s_{\frac{1}{2}}-4p_{\frac{3}{2}}$	393.366
$4s_{\frac{1}{2}}-3d_{\frac{3}{2}},m=\frac{1}{2}$
		1307.60(96.2)	1252.44	1308.590(71.108)
$3d_{\frac{3}{2}}-4p_{\frac{1}{2}}$	866.214
		850.3301(18)	850.33	850.335(2)
$3d_{\frac{3}{2}}-4p_{\frac{3}{2}}$	849.802
$4s_{\frac{1}{2}}-4p_{\frac{1}{2}}$	396.847
		395.7962(1)	395.80	395.7981(1)
$4s_{\frac{1}{2}}-4p_{\frac{3}{2}}$	393.366
$4s_{\frac{1}{2}}-3d_{\frac{5}{2}},m=\frac{5}{2}$
$4s_{\frac{1}{2}}-4p_{\frac{1}{2}}$	396.847
		395.7949(1)	395.79	395.7968(1)
$4s_{\frac{1}{2}}-4p_{\frac{3}{2}}$	393.366
$4s_{\frac{1}{2}}-3d_{\frac{5}{2}},m=\frac{3}{2}$
		1073.80(31.61)	1052.26	1074.336(26.352)
$3d_{\frac{5}{2}}-4p_{\frac{3}{2}}$	854.209
$4s_{\frac{1}{2}}-4p_{\frac{1}{2}}$	396.847
		395.7958(1)	395.79	395.7978(1)	395.7992(2)
$4s_{\frac{1}{2}}-4p_{\frac{3}{2}}$	393.366
$4s_{\frac{1}{2}}-3d_{\frac{5}{2}},m=\frac{1}{2}$
		1337.30(115.38)	1271.92	1338.474(82.593)
$3d_{\frac{5}{2}}-4p_{\frac{3}{2}}$	854.209
$4s_{\frac{1}{2}}-4p_{\frac{1}{2}}$	396.847
		395.7963(1)	395.79	395.7982(1)	395.7990(2)
$4s_{\frac{1}{2}}-4p_{\frac{3}{2}}$	393.366

Table 4. Table 4: Magic wavelengths (in nm) for the 4 ​ s 1 2 ​ m = − 1 2 − 3 ​ d 3 2 ​ m j 4 subscript 𝑠 1 2 𝑚 1 2 3 subscript 𝑑 3 2 subscript 𝑚 𝑗 4s_{\frac{1}{2}m=-\frac{1}{2}}-3d_{\frac{3}{2}m_{j}} transition of Ca + with the circularly polarized lights.

A=-1	A=1	A=-1	A=1
$m_j=\frac{3}{2}$		$m_j=-\frac{3}{2}$
1584.95(143.70)			1580.01(142.74)
851.1728(42)			851.1724(42)
	394.6315(2)		394.6394(4)
$m_j=\frac{1}{2}$		$m_j=-\frac{1}{2}$
1036.70(20.30)			1035.46(20.15)
853.5998(264)	876.07(2.59)	876.28(2.61)	853.5974(266)
	394.6335(1)		394.6362 (1)

Table 5. Table 5: Magic wavelengths (in nm) for the 4 ​ s 1 2 ​ m = − 1 2 − 3 ​ d 5 2 ​ m ​ j 4 subscript 𝑠 1 2 𝑚 1 2 3 subscript 𝑑 5 2 𝑚 𝑗 4s_{\frac{1}{2}m=-\frac{1}{2}}-3d_{\frac{5}{2}mj} transition of Ca + with the circularly polarized lights.

A=-1	A=1	A=-1	A=1
$m_j=\frac{5}{2}$		$m_j=-\frac{5}{2}$
1732.77(200.04)			1726.68(198.7)
	394.6311(3)		394.6400(4)
$m_j=\frac{3}{2}$		$m_j=-\frac{3}{2}$
1187.42(46.79)			1185.07(46.46)
	394.6324(2)		394.6377(2)
$m_j=\frac{1}{2}$		$m_j=-\frac{1}{2}$
987.60(14.87)	894.50(4.03)	894.81(4.06)	986.63(14.76)
	394.6339(1)		394.6357(1)

Equations

$$\text{H}=c\mathbf{\alpha }\cdot \mathbf{p}+\beta c^2+V_{core}(\mathbf{r}).$$

$$V_{core}(\mathbf{r})=-\frac{Z}{r}+V_{dir}(\mathbf{r})+V_{exc}(\mathbf{r})+V_p(\mathbf{r}),$$

$$V_p(r)=\sum _{\ell m}\frac{\alpha g_{\ell }^2(r)}{2r^4}|\ell m⟩⟨\ell m|,$$

$${\alpha }_i(\omega )$$

$${\alpha }_i(\omega )={\alpha }_i^0(\omega )+\frac{3m_{j_i}^2-j_i(j_i+1)}{j_i(2j_i-1)}{\alpha }_i^2(\omega ).$$

$${\alpha }_i(\omega )={\alpha }_i^0(\omega )+A\frac{m_{j_i}}{2j_i}{\alpha }_i^1(\omega )-\frac{3m_{j_i}^2-j_i(j_i+1)}{2j_i(2j_i-1)}{\alpha }_i^2(\omega ).$$

$${\alpha }_i^0(\omega )$$

$${\alpha }_i^1(\omega )$$

$$\times$$

$${\alpha }_i^2(\omega )$$

$$\times$$

$$\Delta \nu =-\frac{1}{2}(831.9V/m{)}^2(\frac{T(K)}{300}{)}^4{\alpha }_i^0(1+\eta ),$$

$$\eta$$

$$\Delta {\nu }_{BBR}(4s-3d_{\frac{5}{2}})=\Delta \nu (3d_{\frac{5}{2}})-\Delta \nu (4s).$$

$$\underset{\omega \to \Delta E(4s-n{p}_{\frac{1}{2}})}{\lim }(\frac{1}{3(2j_i+1)}|⟨{\psi }_{4s}\Vert D\Vert {\psi }_{n{p}_{\frac{1}{2}}}⟩{|}^2$$

$$=$$

$$\underset{\omega \to \Delta E(3d_{\frac{3}{2}}-n{p}_j)}{\lim }(\frac{1}{3(2j_i+1)}|⟨{\psi }_{3d_{\frac{3}{2}}}\Vert D\Vert {\psi }_{n{p}_j}⟩{|}^2$$

$$\times$$

$$=$$

Full text

I INTRODUCTION

The magic wavelength, at which the ac Stark shift of the transition energy is zero for the certain frequencies, was introduced in Refs.Ye et al. (1999); Katori et al. (1999). The magic wavelengths have been extensively used in ultraprecise optical lattice clocksTakamoto and Katori (2003); Bauch (2003); Gill et al. (2003); Gill (2005); Lorini et al. (2008); Gill (2011); Kirchmair et al. (2009), the state-insensitive quantum engineering Sahoo and Arora (2013); Wilpers et al. (2007).

The magic wavelengths for the linearly polarized light have been studied for alkali-metal and alkaline-earth-metal atoms in experiment and theoryLudlow et al. (2008); Kaur et al. (2015); Liu et al. (2015); Tang et al. (2013); Lundblad et al. (2010); Arora et al. (2007); Singh et al. (2016a). The magic wavelengths of alkali-metal atoms for the circularly polarized light have been calculatedArora and Sahoo (2012); Sahoo and Arora (2013); Singh et al. (2016b, c). The use of circularly polarized light has advantages owing to the vector polarizabilities which are absent in the linearly polarized light, such as magnetic-sublevel selective trapping and far-off-resonance laser trappingLe Kien et al. (2013); Sahoo and Arora (2013).

Ca+ is an alkali-metal like ion. Since the nuclear spin is zero for 40Ca*+, it is immune to the first-order Zeeman frequency shiftGao (2013) and convenient for laser cooling. The 40Ca+* ion is preferred for optical frequency standard and quantum computingHuang et al. (2012); Degenhardt et al. (2004); Chwalla et al. (2009); Zhang et al. (2017); Häffner et al. (2008). The frequency of 40Ca*+* optical clocks has been measured with an uncertainty at the $10^{-17}$ level Huang et al. (2016).

Recently, two magic wavelengths of the 40Ca*+* $4s-3d_{\frac{5}{2},m=\frac{1}{2},\frac{3}{2}}$ clock transitions for the linearly polarized light are measured with very high precision Liu et al. (2015). This measurement agrees with the theoretical predictions of B-spline Dirac-Fock plus core polarization (DFCP) method Tang et al. (2013) excellently. Meanwhile, the measurement for these two magic wavelengths determines the ratio of $4s-4p_{\frac{1}{2}}$ and $4s-4p_{\frac{3}{2}}$ oscillator strength with the deviation less than 0.5%.

In this paper, the energy levels, electric dipole matrix elements and static polarizabilities are calculated using relativistic configuration interaction plus core polarization (RCICP) method. The dynamic dipole polarizabilities of the $4s$, $4p_{j}$ and $3d_{j}$ states of Ca*+* ion are calculated for the linearly and circularly polarized light. The magic wavelengths for each of magnetic sublevel transitions are determined. In Sec. II., a brief description of the theoretical method is presented. In Sec. III., the static and dynamic polarizabilities and magic wavelengths are discussed. In Sec. IV., a few conclusions are pointed out. The unit used in the present calculations is atomic unit (a.u.).

II Theoretical method

The energy levels and transition arrays are calculated using RCICP method which has been developed recently Jiang et al. (2016). The present method is similar to the calculation of magic wavelengths of Ca+ for the linearly polarized light by Tang et al. Tang et al. (2013), except that they use the B-spline basis. The basic strategy of the model is to partition the atom into valence and core electrons. The first step involves a Dirac-Fock (DF) calculation of the Ca2+ ground state. The orbitals of the core are written as linear combinations of S-spinors which can be treated as relativistic generalizations of the Slater-type orbitals.

Then, the effective interaction of the valence electron with the core is written as

[TABLE]

The core operator is

[TABLE]

where $V_{dir}$, $V_{exp}$ are direct and exchange interactions with core electrons. The semi-empirical core polarization potential is introduced to approximate the correlation interaction between the core and valence electrons. The $\ell$-dependent polarization potential $V_{p}$ can be written as

[TABLE]

$\alpha$ is the static dipole polarizability of the core, $\alpha$ = 3.26 a.u.Safronova and Safronova (2011b) and $g_{\ell}^{2}(r)$ is a cutoff function, $g_{\ell}^{2}(r)=1-exp(-r^{6}/\rho_{\ell}^{6})$. $\rho_{\ell}$ is an adjustable parameter that is tuned to reproduce the binding energies of the corresponding states. The orbitals of the valence electron are written as linear combinations of L-spinors and S-spinors. L-spinors can be treated as relativistic generalizations of the Laguerre -type orbitals. See Supplemental Tables I and II for lists of energy levels and electric-dipole matrix elements for some low-lying excited states transitions of Ca*+* ions sup .

For arbitrary polarized light, the dynamic polarizability $\alpha_{i}(\omega)$ is given by Le Kien et al. (2013); Manakov et al. (1986); Beloy (2009)

[TABLE]

where $\alpha_{i}^{0}(\omega)$, $\alpha_{i}^{1}(\omega)$, $\alpha_{i}^{2}(\omega)$ are the scalar, vector, and tensor polarizabilities for state $i$, respectively; $j$ ,$m_{j_{i}}$ are the total angular momentum and the corresponding magnetic quantum number. $\theta_{k}$ is the angle between the wave vector of the electric field and $z$-axis. $\theta_{p}$ relates to the direction of polarization vector and quantization axis which was defined in Ref.Le Kien et al. (2013); Manakov et al. (1986). From geometrical considerations, it is found that $\theta_{k}$ and $\theta_{p}$ must satisfy the inequality $cos^{2}\theta_{k}+cos^{2}\theta_{p}\leq 1$. A represents the degree of polarization, which can be taken arbitrary value from $-1$ to $1$. For the linearly polarized light, A is equal to zero. The direction of $z$-axis is polarization direction which is perpendicular to the wave vector of the electric field, $cos\theta_{k}$ = 0 and $cos\theta_{p}$ = 1.

[TABLE]

$A=1$ is for the right handed and $A=-1$ is for the left handed circularly polarized light. Here, we can choose the direction of $z$-axis as the wave vector of the electric field and then $cos\theta_{k}$ = 1, $cos^{2}\theta_{p}$ = [math]. Then the eq.(4) can be simplified as

[TABLE]

The scalar polarizability was conventionally expressed as

[TABLE]

The vector polarizability was written as

[TABLE]

The tensor polarizability was expressed as

[TABLE]

where $|\langle\psi_{i}\|D\|\psi_{n}\rangle|$ and $\Delta E_{in}$ are reduced matrix element and excitation energy of transition respectively. For the state with $j<\frac{1}{2}$, the tensor polarizability makes no contribution to total polarizability.

III POLARIZABILITIES

III.1 STATIC POLARIZABILITIES

When the frequency of light is zero, the dynamic polarizability becomes static dipole polarizability, in which the contribution of vector polarizability is zero. Table 1 lists dipole scalar and tensor static polarizabilities of the low-lying states of Ca*+*, which are compared with the available theoretical and experimental results.

For the ground state, the present polarizabilities are in good agreement with the previous calculationsTang et al. (2013); Mitroy et al. (2008); Chang (1983); Safronova and Safronova (2011a). The errors are within 0.8% except for the value calculated by relativistic coupled cluster (RCC) method. There is a significant difference between the values of RCC and other calculations. The scalar and tensor polarizabilities of the $3d_{j}$ states are excellently agree with the results of Tang et al. Tang et al. (2013). The scalar polarizabilities for the $4p_{j}$ states are negative which are caused by the downward transitions to the lower states.

III.2 BLACKBODY RADIATION SHIFT

The accuracy of optical frequency standards is limited by the frequency shift in the clock transitions caused by the interactions of the ion with external field. The major contributions to the systematic frequency shifts come from blackbody radiation (BBR) shift. The frequency shift for a state due to blackbody radiation at temperature T can be written as

[TABLE]

where $\alpha_{i}^{0}$ is the static scalar polarizability for state $i$. The factor $\eta$ is a small dynamic correction Porsev and Derevianko (2006),

[TABLE]

As Table 2 shows the value of $\eta$ can be negligible. Here $y_{i}=\omega_{in}/T$ and T is temperature, we set as 300 K. The atomic unit for $\alpha$ can be converted to SI units via $\alpha/h[Hz/(V/m)^{2}]=2.48832\times 10^{-8}\alpha[a.u.]$.

The BBR shift for the clock transition is the difference of the BBR shifts between the individual levels involved in the transition and can be written as,

[TABLE]

Table 2 shows the BBR shift and scalar dipole polarizabilities of individual levels for $4s_{\frac{1}{2}}-3d_{\frac{5}{2}}$ clock transition of Ca*+* ion. The present BBR shift is $0.367$ Hz which agrees with the value of Mitroy *et al.*Mitroy et al. (2008) about 0.16%, Sahoo *et al.*Sahoo et al. (2009) about 0.7%, Arora *et al.*Arora et al. (2007) about 3.3%, Champenois *et al.*Champenois et al. (2005) about 5.7% and Kajita *et al.*Kajita et al. (2005) about 8.15%. In the recent experiment, the BBR shift is 0.35(0.009) Hz at temperature T= $294.4\pm 1.6$ KGuan et al. (2016). If we set T = 294.4 K in our calculation, the BBR shift is 0.341 Hz which is within the experimental error bar.

III.3 DYNAMIC POLARIZABILITIES

III.3.1 linearly polarized light

For the case of the linear polarization (A=0), the vector polarizability does not contribute to total polarizability. The polarizability only has scalar component for state with $j=\frac{1}{2}$. The polarizability has scalar and tensor components for state with $j>\frac{1}{2}$. Supplemental Fig.I gives dynamic polarizabilties of $4s$, $4p_{j}$ and $3d_{j}$ states for the linearly polarized lightsup .

Table 3 lists magic wavelengths for the $4s_{\frac{1}{2}}-4p_{\frac{1}{2},\frac{3}{2}}$ and $4s_{\frac{1}{2}}-3d_{\frac{3}{2},\frac{5}{2}}$ transitions of Ca*+* for the linearly polarized light. Uncertainties for all the magic wavelengths have been estimated which are similar to the estimations of Tang *et al.*Tang et al. (2013). For the $4s-4p_{j}$ polarizability difference, the matrix elements of $4s-4p_{j}$, $4p_{j}-5s$, $4p_{j}-3d_{j}$ and $4p_{j}-4d_{j}$ are dominant. For the $4s-3d_{j}$ polarizability difference, the matrix elements of $4s-4p_{j}$ and $3d_{j}-4p_{j}$ are dominant. All these matrix elements were changed by $\pm 2\%$ and the magic wavelengths were recomputed. The magic wavelengths near 368 nm, 395 nm and 850 nm for $4s-4p_{j}$ and $4s-3d_{j}$ transitions agree with the results of available theoretical valuesTang et al. (2013); Kaur et al. (2015) excellently. The maximum difference is 0.7 nm at 850.9217 nm for $4s-4p_{\frac{3}{2}}$ transition. Compared to experimental value Liu et al. (2015), the differences of magic wavelengths at 395.7958 nm and 395.7963 nm for $4s-3d_{\frac{5}{2}}$, are only 0.0034 nm and 0.0027 nm, respectively. Present magic wavelengths near 691 nm, 672 nm, 687 nm, 887 nm, 1073 nm and 1307 nm for $4s-4p_{j}$ and $4s-3d_{j}$ transitions are in good agreement with results of Tang *et al.*Tang et al. (2013). The maximum difference is 0.99 nm near 1307 nm for $4s-3d_{\frac{3}{2}m=\frac{1}{2}}$ transition. But the results of Kaur *et al.*Kaur et al. (2015) have big differences with the present results and Tang’s results Tang et al. (2013) for these magic wavelengths, For example, the differences are 6.4 nm near 691 nm for the $4s_{\frac{1}{2}}-4p_{\frac{1}{2}}$ transition, 55 nm near 1307.60 nm for $4s_{\frac{1}{2}}-3d_{\frac{3}{2}}$ transition. Supplemental Tables III-X lists the breakdown of the polarizabilities at the magic wavelengthssup .

III.3.2 circularly polarized light

For the circularly polarized light, the polarizability has the scalar, vector and tensor components. The dynamic polarizability is different for each of magnetic sublevels of the atomic state. Due to the symmetry, the dynamic polarizabilities of negative $m$ state for $A=-1$ will be same as positive $m$ state for $A=+1$. In the following discussion, we just give the polarizabilites for the ground state $4s_{\frac{1}{2}m=-\frac{1}{2}}$ .

Fig.1 shows the dynamic polarizabilities of the $4s_{\frac{1}{2}m=-\frac{1}{2}}$ and $4p_{\frac{1}{2}m=\pm\frac{1}{2}}$ states for the left and right handed circularly polarized light. In Fig. 1(a), we find that the dynamic polarizabilities of $4s_{\frac{1}{2}m=-\frac{1}{2}}$ state have a big difference with the dynamic polarizabilities for the linearly polarized light. When the wavelength is close to $4s-4p_{\frac{1}{2}}$ transition wavelength, $\alpha_{4s}$ $\rightarrow$ $\infty$ for linearly polarized light. However, the dynamic polarizabilities of $4s_{\frac{1}{2}m=-\frac{1}{2}}$ state are not infinity for left handed circularly polarized light. The reason is that the $4s-4p_{\frac{1}{2}}$ transition has no contribution to the polarizability when the photo energy is close to the $4s-4p_{\frac{1}{2}}$ transition energy. The scalar and vector polarizabilities offset each other, that is

[TABLE]

where, $j_{i}$ = $\frac{1}{2}$ is total angular momentum for 4s state, $j_{n}$ is total angular momentum for $np_{\frac{1}{2}}$ state and $m_{j_{i}}$ = $-\frac{1}{2}$ is magnetic quantum number for $4s$ state. Same situations also happened for $4p_{\frac{1}{2}m=-\frac{1}{2}}$ state when the wavelength is close to $4p_{\frac{1}{2}}-5s$ transition wavelength. There are two magic wavelengths for $4s_{\frac{1}{2}m=-\frac{1}{2}}-4p_{\frac{1}{2}m=-\frac{1}{2}}$ transition for A=$-1$. The first magic wavelength is 395.5410 nm which lies between $4s-4p_{\frac{1}{2}}$ and $4s-4p_{\frac{3}{2}}$ resonances energy. This magic wavelength is very close to the magic wavelength 395.1788 nm for $4s-4p_{\frac{1}{2}}$ transition for linearly polarized light Tang et al. (2013); Kaur et al. (2015). Another magic wavelength 778.37 nm occurs between $4p_{\frac{1}{2}}-4s$ and $4p_{\frac{1}{2}}-3d_{\frac{3}{2}}$ transition energy. This magic wavelength has 87 nm difference with the magic wavelength 691.24 nm of $4s-4p_{\frac{1}{2}}$ transition for linearly polarized light which also lies between $4p_{\frac{1}{2}}-4s$ and $4p_{\frac{1}{2}}-3d_{\frac{3}{2}}$ transitions.

Fig.1(b) gives the dynamic polarizabilities of $4s_{\frac{1}{2}m=-\frac{1}{2}}$ and $4p_{\frac{1}{2}m=-\frac{1}{2}}$ for right handed circularly polarized light. The change of polarizabilities of $4s$ state is similar to linearly polarized light. Three magic wavelengths are found for $4s_{\frac{1}{2}m=-\frac{1}{2}}-4p_{\frac{1}{2}m=-\frac{1}{2}}$ transition. The first magic wavelength 361.83 nm occurs between the energies of the $4p_{\frac{1}{2}}-4d_{\frac{3}{2}}$ and the $4p_{\frac{1}{2}}-5s$ transitions. The second magic wavelength 394.5839 nm occurs between $4s-4p_{\frac{1}{2}}$ and $4s-4p_{\frac{3}{2}}$ resonant transition. The next wavelength is located around 603 nm when the photo energy gets to close to the excitation energies between the $4p_{\frac{1}{2}}-4s$ and $4p_{\frac{1}{2}}-3d_{\frac{3}{2}}$ transitions.

Fig.1(c) gives the dynamic polarizabilities of $4s_{\frac{1}{2}m=-\frac{1}{2}}$ and $4p_{\frac{1}{2}m=+\frac{1}{2}}$ for left handed circularly polarized light. Only two magic wavelengths 362.38 and 600.43 nm are found in the considered range of wavelength. The first one occurs between $4p_{\frac{1}{2}}-4d_{\frac{3}{2}}$ and $4p_{\frac{1}{2}}-5s$ transitions. Another one occurs when the photo energy lies between $4p_{\frac{1}{2}}-3d_{\frac{3}{2}}$ and $4p_{\frac{1}{2}}-4s$ transition energies. There is no magic wavelength between $4s-4p_{\frac{1}{2}}$ and $4s-4p_{\frac{3}{2}}$ resonant transition.

Fig.1(d) gives the dynamic polarizabilities of $4s_{\frac{1}{2}m=-\frac{1}{2}}$ and $4p_{\frac{1}{2}m=+\frac{1}{2}}$ for right handed circularly polarized light. Two magic wavelengths are found for $4s_{\frac{1}{2}m=-\frac{1}{2}}-4p_{\frac{1}{2}m=\frac{1}{2}}$ transition. The first magic wavelength is 394.1570 nm which lies between $4s-4p_{\frac{1}{2}}$ and $4s-4p_{\frac{3}{2}}$ resonant transition. The second magic wavelength is located 779.00 nm when the photo energy lies between the $4p_{\frac{1}{2}}-3d_{\frac{3}{2}}$ and $4p_{\frac{1}{2}}-4s$ transitions.

Fig.2 shows the dynamic polarizabilities of the $4s_{\frac{1}{2}m=-\frac{1}{2}}$ and $4p_{\frac{3}{2}m=\pm\frac{1}{2}}$ states for the left and right handed circularly polarized light. Fig.2(a) shows the dynamic polarizabilities of $4s_{\frac{1}{2}m=-\frac{1}{2}}$ and $4p_{\frac{3}{2}m=\frac{1}{2}}$ states for A=$+1$. There are three magic wavelengths in the range of 360 nm to 1000 nm for this transition. There is no magic wavelength between $4p_{\frac{3}{2}}-4d_{\frac{5}{2}}$ and $4p_{\frac{3}{2}}-5s$ transitions. The first magic wavelength 395.1904 nm lies between the $4s-4p_{\frac{1}{2}}$ and $4s-4p_{\frac{3}{2}}$ resonant transition, that just has 1.0 nm difference compared to that for the linearly polarized light. The second magic wavelength 724.11 nm lies between the $4p_{\frac{1}{2}}-4s$ and $4p_{\frac{3}{2}}-3d_{\frac{3}{2}}$ transitions, which is larger about 37 nm than the 687.51 nm for linearly polarized light. The last one 851.0556 nm is close to the magic wavelength 850.1164 nm for linearly polarized light which lies between the transition energy of $4p_{\frac{3}{2}}-3d_{j}$.

Much more attentions should be paid on the magic wavelength near 851 nm, because this wavelength arises due to cancellations in the polarizabilties from two transitions of $4p_{\frac{3}{2}}\to 3d_{j}$ spin-orbital splitting. From the Table XXV in Suppmlementsup , it can be found that the $4s_{\frac{1}{2}m=-\frac{1}{2}}$ polarizability is dominated by $4s_{\frac{1}{2}}-4p_{j}$ transitions and the $4p_{\frac{3}{2}m=\frac{1}{2}}$ polarizabiltiy is dominated by $4p_{\frac{3}{2}}\to 3d_{j}$ transitions. Combining with the experimental matrix elements of $4s\to 4p_{j}$ transitions, the measurement of this magic wavelength could be able to determine the oscillator strength ratio of $f_{4p_{\frac{3}{2}}\to 3d_{\frac{3}{2}}}:f_{4p_{\frac{3}{2}}\to 3d_{\frac{5}{2}}}$. Suppose that all the remaining components accuracy of $4p_{\frac{3}{2}}$ polarizability including the $4p_{\frac{3}{2}}\to 5s_{\frac{1}{2}}$ and $4d_{j}$ contributions is $5\%$. Then the overall uncertainty to the polarizability is less than $1\%$.

The dynamic polarizabilities of $4s_{\frac{1}{2}m=-\frac{1}{2}}$ and $4p_{\frac{3}{2}m=-\frac{1}{2}}$ states for A=$+1$ are shown in Fig.2(b). There are four magic wavelengths in the range of 360 nm to 1000 nm. The first magic wavelength 370.30 nm occurs when the photo energy lies between energy of $4p_{\frac{3}{2}}-4d_{\frac{5}{2}}$ and $4p_{\frac{3}{2}}-5s$ transitions. This magic wavelength is larger about 0.6 nm than the magic wavelength 369.65 nm for $4s-4p_{\frac{3}{2}m=\frac{1}{2}}$ transition for linearly polarized light, which also lies between $4p_{\frac{3}{2}}-4d_{\frac{5}{2}}$ and $4p_{\frac{3}{2}}-5s$ transitions. The second magic wavelength is 395.5946 nm which lies between the $4s-4p_{\frac{1}{2}}$ and $4s-4p_{\frac{3}{2}}$ resonant transition. The third one is 631.85 nm when photo energy lies between the $4p_{\frac{3}{2}}-3d_{\frac{5}{2}}$ and $4p_{\frac{1}{2}}-4s$ transitions. The last magic wavelength occurs 850.4771 nm which occurs between the $4p_{\frac{3}{2}}-3d_{\frac{3}{2}}$ and $4p_{\frac{3}{2}}-3d_{\frac{5}{2}}$ resonant transition.

The dynamic polarizabilities of $4s_{\frac{1}{2}m=-\frac{1}{2}}$ and $4p_{\frac{3}{2}m=+\frac{1}{2}}$ states for $A=-1$ are shown in Fig.2(c). Three magic wavelengths are got for the $4s_{\frac{1}{2}m=-\frac{1}{2}}-4p_{\frac{3}{2}m=+\frac{1}{2}}$ transition. The first one 370.57 nm lies between the $4p_{\frac{3}{2}}-4d_{\frac{5}{2}}$ and $4p_{\frac{3}{2}}-5s$ transitions. The second one 629.33 nm lies between the $4p_{\frac{3}{2}}-3d_{\frac{5}{2}}$ and $4p_{\frac{1}{2}}-4s$ transitions, that has about 58 nm difference compared to 687.51 nm for the linearly polarized light. The last one is 850.4770 nm which lies between $4p_{\frac{3}{2}}-4d_{j}$ transitions.

The dynamic polarizabilities of $4s_{\frac{1}{2}m=-\frac{1}{2}}$ and $4p_{\frac{3}{2}m=-\frac{1}{2}}$ states for A=$-1$ are shown in Fig.2(d). The only two magic wavelengths are found. The first one 722.97nm lies between the $4p_{\frac{3}{2}}-3d_{\frac{5}{2}}$ and $4p_{\frac{3}{2}}-4s$ transitions. Another one 851.0555 nm occurs between the $4p_{\frac{3}{2}}-3d_{\frac{5}{2}}$ and $4p_{\frac{3}{2}}-3d_{\frac{3}{2}}$ transitions.

Fig.3 shows the dynamic polarizabilities of the $4s_{\frac{1}{2}m=-\frac{1}{2}}$ and $4p_{\frac{3}{2}m=\pm\frac{3}{2}}$ states for the left and right handed circularly polarized light. The dynamic polarizabilities of $4s_{\frac{1}{2}m=-\frac{1}{2}}$ and $4p_{\frac{3}{2}m=+\frac{3}{2}}$ states for A=+1 are shown in Fig.3(a). There are three magic wavelengths in the range of 370 nm to 1000 nm. The first magic wavelength 395.7744 nm occurs when the photo energy lies between energy of $4s-4p_{j}$ resonant transitions. The second magic wavelength 804.28 nm which lies between $4p_{\frac{3}{2}}-4s$ and $4p_{\frac{3}{2}}-3d_{\frac{3}{2}}$ transitions is larger about 131 nm than 672.89 nm for the linearly polarized light. The third magic wavelength 851.8770 nm is close to the magic wavelength 850.9217 nm for the linearly polarized light. Three magic wavelengths are found for the $4s_{\frac{1}{2}m=-\frac{1}{2}}-4p_{\frac{3}{2}m=-\frac{3}{2}}$ transition for the right handed polarized light in Fig.3(b), that are 358.21 nm 394.5596 nm and 563.06 nm. The dynamic polarizabilities of $4s_{\frac{1}{2}m=-\frac{1}{2}}$ and $4p_{\frac{3}{2}m=+\frac{3}{2}}$ states for A=-1 are shown in Fig.3(c). Two magic wavelengths for this transition are found at 359.25 nm and 559.31 nm. The dynamic polarizabilities of $4s_{\frac{1}{2}m=-\frac{1}{2}}$ and $4p_{\frac{3}{2}m=-\frac{3}{2}}$ states for A=-1 are shown in Fig.3(d). Two magic wavelengths 803.96 nm and 851.8764 nm are found.

The dynamic polarizabilities of the $4s_{\frac{1}{2}m=-\frac{1}{2}}$ and $3d_{\frac{3}{2}m}$ states are shown in Fig.4. We also find that the dynamic polarizabilities of $3d_{\frac{3}{2}m=\frac{3}{2}}$ state have a big difference with dynamic polarizabilities for the linearly polarized light. When the wavelength is close to $3d_{\frac{3}{2}}-4p_{\frac{3}{2}}$ and $3d_{\frac{3}{2}}-4p_{\frac{1}{2}}$ transition wavelengths, $\alpha_{3d_{\frac{3}{2}m=\frac{3}{2}}}\to\infty$ for linearly polarized light. The dynamic polarizabilities of $3d_{\frac{3}{2}m=\frac{3}{2}}$ state are not infinity for the right handed circularly polarized light. The $3d_{\frac{3}{2}}-4p_{\frac{1}{2}}$ or $3d_{\frac{3}{2}}-4p_{\frac{3}{2}}$ transition have no contribution to the polarizability when the photo energy is close to their transition energy. The scalar, vector and tensor polarizabilities offset each other,

[TABLE]

where, $j_{i}$ = $\frac{3}{2}$ is total angular momentum for $3d_{\frac{3}{2}}$ state, $j_{n}$ = $\frac{1}{2},\frac{3}{2}$ is total angular momentum for $np_{j}$ state and $m_{j_{i}}$ = $+\frac{3}{2}$ is magnetic quantum number for $3d_{\frac{3}{2}}$ state. The magic wavelengths are listed in Table 4. It can be found that there are two or three magic wavelength for each of $4s_{\frac{1}{2}m=-\frac{1}{2}}-3d_{\frac{3}{2},m}$ transitions. It should be noted that wavelength near 394.6 nm lies between the $4s-4p_{j}$ resonant transitions which is smaller about 1 nm than 394.79 nm of $4s-3d_{\frac{3}{2}m}$ transition for the linearly polarized light. The measurement of magic wavelength near 394.6 nm can be use to a further check the ratio of the $f_{4s-4p_{j}}$ oscillator strength. As mentioned before, the measurement of magic wavelength near 850 nm for $4s_{\frac{1}{2}m=-\frac{1}{2}}-3d_{\frac{3}{2},m=\pm\frac{1}{2}}$ is able to determine the oscillator strength ratio of $f_{4p_{\frac{3}{2}}\to 3d_{\frac{3}{2}}}:f_{4p_{\frac{3}{2}}\to 3d_{\frac{5}{2}}}$.

The dynamic polarizabilities of the $4s_{\frac{1}{2}m=-\frac{1}{2}}$ and $3d_{\frac{5}{2},m}$ states are shown in Fig.5. Table 5 gives the magic wavelengths of $4s_{\frac{1}{2}m=-\frac{1}{2}}-3d_{\frac{5}{2},m}$ transitions. There are only one or two magic wavelength for each of $4s_{\frac{1}{2}m=-\frac{1}{2}}-3d_{\frac{3}{2},m}$ transition. Similar to the experiment of Liu *et al.*Liu et al. (2015), the measurement of magic wavelength near 394.6 nm can be use to a further check the ratio of the $f_{4s-4p_{j}}$ oscillator strength.

IV Conclusions

The energy levels, electric dipole matrix elements and static polarizabilities are calculated using RCICP procedure. The polarizabilties for the $4s$, $4p_{j}$ and $3d_{j}$ states agree with the available theoretical and experimental results very well. The BBR shift for the $4s-3d_{\frac{5}{2}}$ clock transition is also determined and the present result is within the recent experimental error barGuan et al. (2016).

The dynamic dipole polarizabilities of the $4s$, $4p_{j}$ and $3d_{j}$ states of Ca*+* ion are calculated for the linearly and circularly polarized light. The magic wavelengths for each of magnetic sublevels of $4s-4p_{j}$ and $4s-3d_{j}$ transitions are determined. The magic wavelengths for the linearly polarized light agree with the available theoretical resultsTang et al. (2013); Kaur et al. (2015) excellently. The magic wavelengths for the circularly polarized light have very big difference with those for linearly polarized light. Some additional magic wavelength are found for circularly polarized light. We recommend that the measurement of the magic wavelength near 850 nm of $4s-4p_{\frac{3}{2},m=\pm\frac{3}{2},\pm\frac{1}{2}}$ could be able to determine the oscillator strength ratio of $f_{4p_{\frac{3}{2}}\to 3d_{\frac{3}{2}}}$ and $f_{4p_{\frac{3}{2}}\to 3d_{\frac{5}{2}}}$.

V Acknowledgments

The work of JJ was supported by National Natural Science Foundation of China (NSFC) (Grants No.11564036). The work of LYX was supported by NSFC(Grants No. U1331122). The work of DHZ was supported by NSFC (Grants No. 11464042,U1330117). The work of CZD was supported by NSFC (Grants No.11274254, U1332206).