Precision Spectroscopy of Radiation Transitions between Singlet Rydberg States of the Group IIb and Yb Atoms

Abstract

The measurements of microwave (μw) and radio-frequency (RF) radiation quantitative parameters may be based on the quantum–optical approach to determine the spectral characteristics of radiation transitions between the Rydberg states of atoms. Frequencies and matrix elements are calculated for dipole transitions between opposite-parity Rydberg states $n{{}_{ }{}^1\mathrm{L}}_{\mathrm{L}}$ and $n^{\prime }{{}_{ }{}^1\left(\mathrm{L}\pm 1\right)}_{\mathrm{L}\pm 1}$ (where $n^{\prime }=$ $n,n\pm 1,n\pm 2$) of the singlet series in the alkaline–earth–metal-like atoms of group IIb (Zn, Cd, Hg) and Yb. The matrix elements determine the shifts of Rydberg-state energy levels in the field of resonance μw or RF radiation, splitting the resonance of electromagnetically induced transparency (EIT) for intensely absorbed probe radiation. Numerical computations based on the single-electron quantum defect method (QDM) and the Fues’ model potential (FMP) approach with the use of the most reliable data from the current literature on quantum defect values are performed for frequencies and matrix elements of transitions between singlet Rydberg states of ¹S₀-, ¹P₁-, ¹D₂-, and ¹F₃-series in Zn, Cd, Hg, and Yb atoms. The calculated data are approximated by polynomials in the powers of the principal quantum numbers. The polynomial coefficients are determined with the use of a standard curve-fitting interpolation polynomial procedure for numerically calculated functions. These approximation expressions provide new possibilities for accurately evaluating the frequencies and matrix elements of dipole transitions between Rydberg states over a wide range of quantum numbers n >> 1, accompanied by the emission and absorption of μw and RF photons.

1. Introduction

The use of atomic quantum properties and characteristics is currently the most reliable approach to developing the highest precision set of metrological standards. Time-frequency standards, based on neutral atoms, represent the most spectacular examples of metrological standards with record fractional uncertainties below 10⁻¹⁸, continuously attracting significant attention from researchers [1,2,3].

The splitting of the electromagnetically induced transparency (EIT) resonance in atoms in the field of infrared (IR), microwave (μw), or radio-frequency (RF) radiation may be used as a method for high-precision determination of the quantitative characteristics of electric fields in the indicated spectral ranges [4,5,6,7,8,9]. The frequencies of the radiation transitions between the single-electron Rydberg states of atoms are located exactly in these ranges. Therefore, the alkali-metal atoms with their single valence electrons, which may be easily excited to their Rydberg states without disturbing electrons from the inner shells, were the first to attract the attention of researchers involved in the development of atomic standards for the μw electric fields. The basic atomic characteristics in these studies were the amplitudes and frequencies of the radiation transitions between the Rydberg states. Detailed calculations of these characteristics for alkali-metal atoms (Li, Na, K, Rb, and Cs) were performed in [10]. Similar results for the alkaline–earth–metal atoms of group IIa (Mg, Ca, Sr, and Ba) were obtained in [11]. In this paper, we present similar calculations for the group IIb (Zn, Cd, and Hg) and Yb atoms. Our aim is to derive simple equations for determining the numerical data for the frequencies and amplitudes of radiation transitions between singlet Rydberg states.

Currently, existing databases on the energies of atomic bound states may serve as a source for determining the frequencies of radiation transitions between the Rydberg states of atoms. Modern laser systems provide access and the detection of highly excited states with the use of methods of multiphoton transition spectroscopy [4,12]. The states $n{{}_{ }{}^1\mathrm{F}}_3$, $n{{}_{ }{}^1\mathrm{D}}_2$, $n{{}_{}{}^1\mathrm{P}}_1$, and $n{{}_{ }{}^1\mathrm{S}}_0$ are the most suitable for observing μw transitions in alkaline–earth–metal-like atoms because, in close vicinity of their energies, there exist states $n^{\prime }{{}_{}{}^1{\mathrm{L}}^{\prime }}_{{\mathrm{L}}^{\prime }}$ dipole transitions, which are located in Tera-, Giga-, and Mega-Hertz frequency ranges.

In the most reliable databases, the numerical values of energy levels for the nS-, nP-, nD- and nF-series are given only for a finite number of states with $n\le n_{\mathrm{m}\mathrm{a}\mathrm{x}}$, where $n_{\mathrm{m}\mathrm{a}\mathrm{x}}$ depends essentially on the orbital momentum of the series presented in a particular database for a specific atom [13,14]. In particular, for Zn and Hg atoms, the total number of tabulated energy levels in the database [14] is approximately 1.5 times the number $n_{\mathrm{m}\mathrm{a}\mathrm{x}}$ of the base [13]. For Cd and Yb atoms, the total number of tabulated items in the database [14] is approximately three times the corresponding number of the base [13]. Therefore, the numbers $n_{\mathrm{m}\mathrm{a}\mathrm{x}}$ for a particular series of states may also differ in bases similar to the indicated proportions. However, due to the limited digit numbers used for the numerical presentation of energy levels in the data tables, the uncertainty of the numerical values for the parameters determining transition frequencies and matrix elements gradually decreases with increasing corresponding Rydberg-state principal quantum numbers n. Therefore, the values n < 40 are usually sufficient for determining the wave function and transition–energy parameters with a precision of four to five digits.

The frequencies of the most intensely absorbed lines of the alkaline–earth–metal-like atoms are determined by the energy of transitions from their ground states $n_0{{}_{ }{}^1\mathrm{S}}_0$ to the first excited singlet states $n_0{{}_{ }{}^1\mathrm{P}}_1$, where $n_0=$ 4, 5, 6, and 6 are the principal quantum numbers of group IIb (Zn, Cd, Hg) and Yb atoms, respectively. The EIT effect on the probe radiation with the frequency ${\omega }_{\mathrm{p}}=E_{n_0{{}_{ }{}^1\mathrm{P}}_1}-E_{n_0{{}_{}{}^1\mathrm{S}}_0}$ in the atomic vapor appears under the action of sufficiently strong laser radiation, coupling the excited state $n_0{{}_{ }{}^1\mathrm{P}}_1$ to a highly excited Rydberg state $n{{}_{}{}^1\mathrm{L}}_{\mathrm{L}}$ with a principal quantum number $n\gg n_0$. The subscript at the angular momentum L determines the quantum number J of the total orbital momentum J = L + S, which for spin-less singlet states S = 0, coincides exactly with the angular momentum J = L. In the case of a single-photon coupling, the Rydberg-state angular momentum equals 0 or 2. If the Rydberg-state excitation requires N ≥ 1 photon, for an even number of N = 2, 4, …, the Rydberg-state angular momentum may take only odd values from L = 1 to L = N + 1. For odd photon numbers N = 1, 3, …, the angular momentum takes even values from L = 0 to L = N + 1 = 2, 4, … [12]. The photons may be identical, coming from one and the same coupling laser beam of the frequency ${\omega }_i\equiv {\omega }_{\mathrm{c}}$, $i=1,2,\dots N$, or different, coming from N different coupling beams of the frequencies ${\omega }_i\ne {\omega }_k$, $i\ne k=1,2,\dots N$. The resonance of the EIT effect appears when the sum of all coupling laser frequencies ${\omega }_{\Sigma }={\sum }_{i=1}^N{\omega }_i$ coincides exactly with the frequency of transition from the opaque excited state $n_0{{}_{}{}^1\mathrm{P}}_1$ to the Rydberg state $n{{}_{}{}^1\mathrm{L}}_{\mathrm{L}},$ i.e., ${\omega }_{\Sigma }\equiv E_{n{{}_{}{}^1\mathrm{L}}_{\mathrm{L}}}-E_{n_0{{}_{}{}^1\mathrm{P}}_1}=E_{n{{}_{}{}^1\mathrm{L}}_{\mathrm{L}}}-E_{n_0{{}_{}{}^1\mathrm{S}}_0}-{\omega }_{\mathrm{p}}$. It is important to note that, in the case of multiple coupling beams, there may be both absorbed and emitted photons, and then the corresponding frequencies ${\omega }_i$ in the sum ${\omega }_{\Sigma }$ are positive for the former and negative for the latter. The splitting of the EIT resonance using μw radiation may be used for determining the basic characteristics of Rydberg states in atoms, frequencies, and amplitudes of dipole transitions between close Rydberg states.

If, together with the coupling laser field, a μw radiation (IR or RF radiation, for which in what follows we use one and the same notation μw) is applied with a frequency ${\omega }_{\mathsf{\mu }\mathrm{w}}\equiv E_n-E_{n^{\prime }}-\epsilon$ close to the frequency of the $n\to n^{\prime }$ transition ($\left|\epsilon \right|\ll {\omega }_{\mathsf{\mu }\mathrm{w}}$) between Rydberg states $|n{{}_{}{}^1\mathrm{L}}_{\mathrm{L}}⟩$ and $|n^{\prime }{{}_{}{}^1{\mathrm{L}}^{\prime }}_{{\mathrm{L}}^{\prime }}⟩$ with ${\mathrm{L}}^{\prime }=\mathrm{L}\pm 1$, $n^{\prime }=n,n\pm 1,n\pm 2$, then the state $n{{}_{}{}^1\mathrm{L}}_{\mathrm{L}}$ transforms into two possible superpositions of states due to the resonance Stark effect (the Autler–Towns effect [15]). The essential details and general equations for this effect are presented in Section 2.

The frequencies and corresponding matrix elements are determined for the μw dipole transitions between the singlet Rydberg states of the alkaline–earth–metal-like atoms of group IIb elements (Zn, Cd, and Hg) and Yb. The most reliable data on the energy spectra and quantum defects of the n¹S₀-, n¹P₁-, n¹D₂-, and n¹F₃-series of bound states are used to evaluate the frequencies of the μw transitions between the Rydberg states. The numerical results of the calculations and their extrapolations to states with extremely large principal quantum numbers n are presented in Section 3. The values of the μw transition matrix elements $\mathcal{R}$ are calculated in the single-electron approximation using the Fues’ model potential (FMP) and the quantum defect method (QDM). The numerical results and quadratic polynomial approximations for evaluating the amplitudes of transitions between Rydberg states are presented in Section 4. In Section 5, the results of numerical calculations are discussed.

2. μw-Radiation-Induced Splitting of EIT Resonance

The energies of the Rydberg-state superpositions in the field of resonant μw radiation are determined as solutions to the secular equation for degenerate quasienergy states the following [16,17,18]:

$$E_n^{\pm }=E_{n{{}_{}{}^1\mathrm{L}}_{\mathrm{L}}}\pm ∆E\left(\epsilon ,\mathsf{\Omega }\right),$$

(1)

with the Autler–Towns resonance splitting

$$∆E\left(\epsilon ,\mathsf{\Omega }\right)=\frac{1}{2}\sqrt{{\epsilon }^2+{\mathsf{\Omega }}^2},$$

(2)

where $\mathsf{\Omega }=F\mathcal{R}$ is the amplitude of the μw transition between Rydberg states $|n⟩$ and $|n^{\prime }⟩$ (the “Rabi frequency”), F is the μw electric field, $\mathcal{R}=⟨n^{\prime }|z|n⟩$ is the matrix element of the electric dipole moment z-component (hereafter, the atomic system of units used is $e=m=\hslash =1$).

For the EIT resonance condition,

$${\omega }_{\mathrm{p}}+{\omega }_{\Sigma }=E_{n{{}_{}{}^1\mathrm{L}}_{\mathrm{L}}}-E_{n_0{{}_{}{}^1\mathrm{S}}_0},$$

(3)

the μw-induced splitting of the Rydberg-state energy (1), and (2) will evidently result in the EIT resonance splitting for the probe radiation ${\omega }_{\mathrm{p}}\to {\omega }_{\mathrm{p}}^{\pm }+∆{\omega }^{\pm }$. Then, the relation (3) transforms into resonance conditions for two separate resonances as follows:

$${\omega }_{\mathrm{p}}^{+}+{\omega }_{\Sigma }=E_n^{+}-E_{n_0{{}_{ }{}^1\mathrm{S}}_0}, \mathrm{a}\mathrm{n}\mathrm{d} {\omega }_{\mathrm{p}}^{-}+{\omega }_{\Sigma }=E_n^{-}-E_{n_0{{}_{ }{}^1\mathrm{S}}_0},$$

(4)

where $E_n^{\pm }$ are determined in (1) and (2). The difference between these relations determines the splitting of the EIT resonances $∆{\omega }_{\mathrm{p}}={\omega }_{\mathrm{p}}^{+}-{\omega }_{\mathrm{p}}^{-}=2∆E\left(\epsilon ,\mathsf{\Omega }\right)$ observed at the probe-wave frequencies ${\omega }_{\mathrm{p}}^{\pm }={\omega }_{\mathrm{p}}\pm ∆E\left(\epsilon ,\mathsf{\Omega }\right)$. It is necessary to note that in the case of collinear propagation of the probe and coupling beams, the linear Doppler effect on the frequencies ${\omega }_{\mathrm{p}}$ and ${\omega }_{\Sigma }$ is one and the same for both frequencies ${\omega }_{\mathrm{p}}^{+}$ and ${\omega }_{\mathrm{p}}^{-}$; therefore, it cancels out their difference $∆{\omega }_{\mathrm{p}}$ [10,11]. Thus, the probe-wave resonance splitting coincides exactly with twice the resonance shift (2), enabling the determination of the μw radiation electric field F from the EIT resonance splitting. The total number of Rydberg states in atoms is practically infinite, whereas the selection of probe ${\omega }_{\mathrm{p}}$ and coupling ${\omega }_{\Sigma }$ laser frequencies providing the multi-photon transition from the ground state to the Rydberg $n{{}_{}{}^1\mathrm{L}}_{\mathrm{L}}$ state is not too difficult [4,12]. Therefore, the measurements of any μw radiation field F may always be realized by choosing four bound states (ground $n_0{{}_{ }{}^1\mathrm{S}}_0$, resonance $n_0{{}_{}{}^1\mathrm{P}}_1$, and a pair of Rydberg states $n$ and $n^{\prime }$) together with a number of resonant coupling beams. For an exact coincidence of the frequency ${\omega }_{\mathsf{\mu }\mathrm{w}}$ with the frequency of the transition between Rydberg states (that is, for $\epsilon =0$ in $∆E\left(\epsilon ,\mathsf{\Omega }\right)$ of Equation (2)), the splitting $∆{\omega }_{\mathrm{p}}$ coincides with the Rabi frequency $\mathsf{\Omega }$. Then, the electric field of μw radiation with a frequency exactly equal to the transition frequency ${\omega }_{\mathsf{\mu }\mathrm{w}}=\left|E_n-E_{n^{\prime }}\right|$ is as follows:

$$F=\mathsf{\Omega }/\mathcal{R},$$

(5)

This relation holds for all atoms in an atomic vapour, independent of the vapour temperature, if all laser beams (probe and coupling) co- or counter-propagate along one and the same line with coinciding directions of their linearly polarized electric field vectors [6,7,8,9,10,11,17] (see Figure 1a). So, for the counter-propagating probe and coupling waves, the Doppler-effect-insensitive difference $∆{\omega }_{\mathrm{p}}$ remains equal to $2∆E\left(\epsilon ,\mathsf{\Omega }\right)$ for all atoms in the laboratory reference frame, independent of the atomic thermal velocities.

Thus, for a given frequency of μw radiation in the frequency ranges of MHz, GHz up to a few THz, Rydberg state $n{{}_{}{}^1\mathrm{L}}_{\mathrm{L}}$ should be determined with a close $n^{\prime }{{}_{}{}^1{\mathrm{L}}^{\prime }}_{{\mathrm{L}}^{\prime }}$ state (${\mathrm{L}}^{\prime }=\mathrm{L}\pm 1$), providing equality ${\omega }_{\mathsf{\mu }\mathrm{w}}=\left|E_n-E_{n^{\prime }}\right|$. After calculation of the matrix element $\mathcal{R}=⟨n^{\prime }|z|n⟩$ the μw electric field may be evaluated using Equation (5), where the experimentally determined values of the EIT resonance splitting $∆{\omega }_{\mathrm{p}}$ should substitute the Rabi frequency $\mathsf{\Omega }$.

3. Frequencies of μw Transitions between Rydberg States of Alkaline–Earth–Metal-like Atoms

The difference between the energy of $n^{\prime }{{}_{}{}^1{\mathrm{L}}^{\prime }}_{{\mathrm{L}}^{\prime }}$ and $n{{}_{}{}^1\mathrm{L}}_{\mathrm{L}}$ levels for ${\mathrm{L}}^{\prime }=\mathrm{L}\pm 1$ and $\left|n^{\prime }-n\right|\le 2$ vanishes as 1/n³ with an increase in the principal quantum number n. Therefore, the determination of transition energy from tables of energy levels becomes impossible already for n > 20, since the terms in the relations $∆E_{n\mathrm{L}{\mathrm{L}}^{\prime }}=E_{n^{\prime }{{}_{}{}^1{\mathrm{L}}^{\prime }}_{{\mathrm{L}}^{\prime }}}-E_{n{{}_{}{}^1\mathrm{L}}_{\mathrm{L}}}$ differ only in the 6th or 7th digit number for $n>15$. Thus, for the most precise determination of $∆E_{n\mathrm{L}{\mathrm{L}}^{\prime }}$ the data for quantum defects of the bound states should be used. The definition of the quantum defect ${\mu }_{n\mathrm{L}}$ with a principal quantum number n and orbital quantum number L is based on the Rydberg equation for the bound-state energy (in the units of cm⁻¹):

$$E_{n\mathrm{L}}=I{p}_A-\frac{{Ry}_A}{{\left(n-{\mu }_{n\mathrm{L}}\right)}^2},$$

(6)

with $I{p}_A$, the energy of a single-electron ionization of an atom from its ground state, ${Ry}_A={Ry}_{\infty }/\left(1+1/M_A\right),$ is the Rydberg constant, taking into account the finite mass $M_A$ of atom A in the units of the electron mass, ${Ry}_{\infty }=\mathrm{109,737.315685}$ cm⁻¹ is the universal Rydberg constant (for $M_A=\infty$) given by the CODATA recommended values of the fundamental constants (https://physics.nist.gov/Constants (accessed on 25 August 2023)).

The quantum defect ${\mu }_{n\mathrm{L}}$ for a series of states with a fixed orbital momentum L is practically independent of the principal quantum number n (${\mu }_{n\mathrm{L}}\approx {\mu }_0$ [19]) in the region of n > 20 and may be presented as a resolution of the form [20,21].

$${\mu }_{n\mathrm{L}}=\sum _{q=0}^{q_{\mathrm{m}\mathrm{a}\mathrm{x}}}\frac{{\mu }_{2q}}{{\left(n-{\mu }_0\right)}^{2q}},$$

(7)

where ${\mu }_{2q}$ ($q=0,1,2,\dots ,q_{\mathrm{m}\mathrm{a}\mathrm{x}}$) are constant parameters for a series of states with fixed values of the total spin S and angular momentum L. In this paper, we considered only singlet states with total spin S = 0. A sufficient number of theoretical and experimental studies were performed to determine the numerical values of constants ${\mu }_{2q}$, $I{p}_A,$ and ${Ry}_A$ [20,21,22,23,24], providing high-precision values of the bound-state energies (6) in alkaline–earth–metal-like atoms of group IIb (A = Zn, Cd, and Hg) and Yb.

In

Table 1. Numerical data for constants of Equations (6) and (7) determining quantum defects and energies of alkaline–earth–metal-like atom singlet Rydberg states.

Series n 1LJ	${\mathit{\mu }}_{2\mathit{q}}$	Atom (and Its Ground State)
		Zn (3d10 4s2 (1S0))	Cd (4d10 5s2 (1S0))	Hg (5d106s2(1S0))	Yb (4f14 6s2 (1S0))
n 1S0	${\mu }_0$	2.642512	3.663305	4.67192	4.28365
	${\mu }_2$	−2.89467	−6.11623	−6.02082	−10.3200
	${\mu }_4$	104.771	108.891	144.630	−70.1351
n 1P1	${\mu }_0$	2.09749	3.0583	4.06980	3.97663
	${\mu }_2$	−1.6302	−2.0660	−8.33250	−6.0866
	${\mu }_4$	86.502	155.708	185.691	162.57
n 1D2	${\mu }_0$	1.231886	2.22767	3.10517	2.721
	${\mu }_2$	−2.51951	−0.932065	−3.3896	−2.0871
	${\mu }_4$	131.023	−39.03163	97.3023	−19.7052
n 1F3	${\mu }_0$	0.02302346	0.0368361	1.05146	1.2894
	${\mu }_2$	0.188324	0.027443	−3.4951	−7.869
	${\mu }_4$	−4.61722	−3.54482	119.614	84.757
RyA (cm−1)		109,736.395	109,736.7802	109,737.0156	109,736.9677
IpA (cm−1)		75,769.33	72,540.07	84,184.15	50,443.0704

, the most reliable values of constants ${\mu }_{2q}$ are presented, taken from the literature and those derived from Equation (6) using the data [13,14] for Rydberg-state energies. From the numerical values of ${\mu }_{2q}$, for n > 15, the main contribution to the sum (7) comes from the two terms $q=0.1.$ We verified the data of Table 1 via a detailed comparison of the energies given by Equation (6) with those of the databases [13,14]. The coincidence of the compared values up to the last digits for all considered atoms confirms the high precision of the calculated results for frequencies and matrix elements presented below.

The numerical values of frequencies are presented In

Table 2. Frequencies of electric dipole transitions between Rydberg states of Zn atoms: from $n{{}_{}{}^1\mathrm{S}}_0$ to $\left(n-1\right){{}_{}{}^1\mathrm{P}}_1$ state, $∆E_{n\mathrm{S}\mathrm{P}}=E_{n{{}_{}{}^1\mathrm{S}}_0}-E_{\left(n-1\right){{}_{}{}^1\mathrm{P}}_1}$; from $n{{}_{}{}^1\mathrm{P}}_1$ to $n{{}_{}{}^1\mathrm{S}}_0$ state, $∆E_{n\mathrm{P}\mathrm{S}}=E_{n{{}_{}{}^1\mathrm{P}}_1}-E_{n{{}_{}{}^1\mathrm{S}}_0};$ from $\left(n+1\right){{}_{}{}^1\mathrm{P}}_1$ to $n{{}_{}{}^1\mathrm{D}}_2$ state, $∆E_{n\mathrm{P}\mathrm{D}}=E_{\left(n+1\right){{}_{}{}^1\mathrm{P}}_1}-E_{n{{}_{}{}^1\mathrm{D}}_2}$; from $n{{}_{}{}^1\mathrm{D}}_2$ to $\left(n-2\right){{}_{}{}^1\mathrm{F}}_3$ state, $∆E_{n\mathrm{D}\mathrm{F}}=E_{n{{}_{}{}^1\mathrm{D}}_2}-E_{\left(n-2\right){{}_{}{}^1\mathrm{F}}_3}$; from $\left(n-1\right){{}_{}{}^1\mathrm{F}}_3$ to $n{{}_{}{}^1\mathrm{D}}_2$ state, $∆E_{n\mathrm{F}\mathrm{D}}=E_{\left(n-1\right){{}_{}{}^1\mathrm{F}}_3}-E_{n{{}_{}{}^1\mathrm{D}}_2}$.

n	$∆{\mathit{E}}_{\mathit{n}\mathbf{S}\mathbf{P}}$, GHz	$∆{\mathit{E}}_{\mathit{n}\mathbf{P}\mathbf{S}}$, GHz	$∆{\mathit{E}}_{\mathit{n}\mathbf{P}\mathbf{D}}$, GHz	$∆{\mathit{E}}_{\mathit{n}\mathbf{D}\mathbf{F}}$, GHz	$∆{\mathit{E}}_{\mathit{n}\mathbf{F}\mathbf{D}}$, GHz
20	600.0652	648.970	130.054	846.8076	197.921
50	28.6296	33.1524	7.57120	46.0590	11.7095
100	3.26784	3.85265	0.915272	5.47010	1.41991
150	0.940038	1.11441	0.268122	1.59392	0.416259
200	0. 390809	0.464549	0.112467	0.666882	0.174659
Parameters of interpolation Equation (8)
$d_0$, THz	3001.46	3588.88	884.234	5207.72	1374.16
$d_1$	8.07501	6.94763	3.50412	4.79019	3.40025
$d_2$	78.2588	39.6976	0.576217	24.5359	−7.10743

,

Table 3. Frequencies of electric dipole transitions between Rydberg states of Cd atoms: from $n{{}_{}{}^1\mathrm{S}}_0$ to $\left(n-1\right){{}_{}{}^1\mathrm{P}}_1$ state, $∆E_{n\mathrm{S}\mathrm{P}}=E_{n{{}_{}{}^1\mathrm{S}}_0}-E_{\left(n-1\right){{}_{}{}^1\mathrm{P}}_1};$ from $n{{}_{}{}^1\mathrm{P}}_1$ to $n{{}_{}{}^1\mathrm{S}}_0$ state, $∆E_{n\mathrm{P}\mathrm{S}}=E_{n{{}_{}{}^1\mathrm{P}}_1}-E_{n{{}_{}{}^1\mathrm{S}}_0};$ from $\left(n+1\right){{}_{}{}^1\mathrm{P}}_1$ to $n{{}_{}{}^1\mathrm{D}}_2$ state, $∆E_{n\mathrm{P}\mathrm{D}}=E_{\left(n+1\right){{}_{}{}^1\mathrm{P}}_1}-E_{n{{}_{}{}^1\mathrm{D}}_2};$ from $n{{}_{}{}^1\mathrm{D}}_2$ to $n{{}_{}{}^1\mathrm{P}}_1$ state, $∆E_{n\mathrm{D}\mathrm{P}}=E_{n{{}_{}{}^1\mathrm{D}}_2}-E_{n{{}_{}{}^1\mathrm{P}}_1}$; from $\left(n-2\right){{}_{}{}^1\mathrm{F}}_3$ to $n{{}_{}{}^1\mathrm{D}}_2$ state, $∆E_{n\mathrm{F}\mathrm{D}}=E_{\left(n-2\right){{}_{}{}^1\mathrm{F}}_3}-E_{n{{}_{}{}^1\mathrm{D}}_2}$.

n	$∆{\mathit{E}}_{\mathit{n}\mathbf{S}\mathbf{P}}$, GHz	$∆{\mathit{E}}_{\mathit{n}\mathbf{P}\mathbf{S}}$, GHz	$∆{\mathit{E}}_{\mathit{n}\mathbf{P}\mathbf{D}}$, GHz	$∆{\mathit{E}}_{\mathit{n}\mathbf{D}\mathbf{P}}$, GHz	$∆{\mathit{E}}_{\mathit{n}\mathbf{F}\mathbf{D}}$, GHz
20	641.336	839.645	197.410	1043.09	216.158
50	26. 5831	39.1125	10.1946	51.4334	11.4226
100	2.92795	4.40750	1.19009	5.92182	1.33881
150	0.833096	1.26206	0.344878	1.70794	0.388282
200	0.344515	0.523459	0.143906	0.710946	0.162065
Parameters of interpolation Equation (8)
$d_0$, THz	2624.22	3987.14	1114.65	5475.17	1255.61
$d_1$	9.60881	9.74238	6.37655	7.56129	6.39761
$d_2$	189.875	79.0349	39.2051	58.4159	22.9400

,

Table 4. Frequencies of electric dipole transitions between Rydberg states of Hg atoms: from $n{{}_{}{}^1\mathrm{S}}_0$ to $\left(n-1\right){{}_{}{}^1\mathrm{P}}_1$ state, $∆E_{n\mathrm{S}\mathrm{P}}=E_{n{{}_{}{}^1\mathrm{S}}_0}-E_{\left(n-1\right){{}_{}{}^1\mathrm{P}}_1};$ from $n{{}_{}{}^1\mathrm{P}}_1$ to $n{{}_{}{}^1\mathrm{S}}_0$ state, $∆E_{n\mathrm{P}\mathrm{S}}=E_{n{{}_{}{}^1\mathrm{P}}_1}-E_{n{{}_{}{}^1\mathrm{S}}_0}$; from $\left(n+1\right){{}_{}{}^1\mathrm{P}}_1$ to $n{{}_{}{}^1\mathrm{D}}_2$ state, $∆E_{n\mathrm{P}\mathrm{D}}=E_{\left(n+1\right){{}_{}{}^1\mathrm{P}}_1}-E_{n{{}_{}{}^1\mathrm{D}}_2};$ from $n{{}_{}{}^1\mathrm{D}}_2$ to $n{{}_{}{}^1\mathrm{P}}_1$ state, $∆E_{n\mathrm{D}\mathrm{P}}=E_{n{{}_{}{}^1\mathrm{D}}_2}-E_{n{{}_{}{}^1\mathrm{P}}_1}$; from $\left(n-2\right){{}_{}{}^1\mathrm{F}}_3$ to $n{{}_{}{}^1\mathrm{D}}_2$ state, $∆E_{n\mathrm{F}\mathrm{D}}=E_{\left(n-2\right){{}_{}{}^1\mathrm{F}}_3}-E_{n{{}_{}{}^1\mathrm{D}}_2}$.

n	$∆{\mathit{E}}_{\mathit{n}\mathbf{S}\mathbf{P}}$, GHz	$∆{\mathit{E}}_{\mathit{n}\mathbf{P}\mathbf{S}}$, GHz	$∆{\mathit{E}}_{\mathit{n}\mathbf{P}\mathbf{D}}$, GHz	$∆{\mathit{E}}_{\mathit{n}\mathbf{D}\mathbf{P}}$, GHz	$∆{\mathit{E}}_{\mathit{n}\mathbf{F}\mathbf{D}}$, GHz
20	721.729	1045.17	69.8134	1404.05	72.8383
50	28. 3622	41.7668	2.39541	63.3287	3.42273
100	3.03030	4.53200	0.259470	7.07836	0.388177
150	0.853440	1.28298	0.0738689	2.02177	0.111430
200	0.351132	0.529215	0.0305908	0.837533	0.0462760
Parameters of interpolation Equation (8)
$d_0$, THz	2624.05	4002.39	239.444	6365.83	354.146
$d_1$	13.3090	11.0088	3.55212	10.1445	8.76692
$d_2$	213.961	215.459	461.965	102.902	82.8157

and

Table 5. Frequencies of electric dipole transitions between Rydberg states of Yb atoms: from $n{{}_{}{}^1\mathrm{S}}_0$ to $\left(n-1\right){{}_{}{}^1\mathrm{P}}_1$ state, $∆E_{n\mathrm{S}\mathrm{P}}=E_{n{{}_{}{}^1\mathrm{S}}_0}-E_{\left(n-1\right){{}_{}{}^1\mathrm{P}}_1};$ from $n{{}_{}{}^1\mathrm{P}}_1$ to $n{{}_{}{}^1\mathrm{S}}_0$ state, $∆E_{n\mathrm{P}\mathrm{S}}=E_{n{{}_{}{}^1\mathrm{P}}_1}-E_{n{{}_{}{}^1\mathrm{S}}_0}$; from $n{{}_{}{}^1\mathrm{D}}_2$ to $\left(n+1\right){{}_{}{}^1\mathrm{P}}_1$ state, $∆E_{n\mathrm{D}\mathrm{P}}=E_{n{{}_{}{}^1\mathrm{D}}_2}-E_{\left(n+1\right){{}_{}{}^1\mathrm{P}}_1};$ from $n{{}_{}{}^1\mathrm{D}}_2$ to $\left(n-2\right){{}_{}{}^1\mathrm{F}}_3$ state, $∆E_{n\mathrm{D}\mathrm{F}}=E_{n{{}_{}{}^1\mathrm{D}}_2}-E_{\left(n-2\right){{}_{}{}^1\mathrm{F}}_3}$; from $\left(n-1\right){{}_{}{}^1\mathrm{F}}_3$ to $n{{}_{}{}^1\mathrm{D}}_2$ state, $∆E_{n\mathrm{F}\mathrm{D}}=E_{\left(n-1\right){{}_{}{}^1\mathrm{F}}_3}-E_{n{{}_{}{}^1\mathrm{D}}_2}$.

n	$∆{\mathit{E}}_{\mathit{n}\mathbf{S}\mathbf{P}}$, GHz	$∆{\mathit{E}}_{\mathit{n}\mathbf{P}\mathbf{S}}$, GHz	$∆{\mathit{E}}_{\mathit{n}\mathbf{D}\mathbf{P}}$, GHz	$∆{\mathit{E}}_{\mathit{n}\mathbf{D}\mathbf{F}}$, GHz	$∆{\mathit{E}}_{\mathit{n}\mathbf{F}\mathbf{D}}$, GHz
20	1283.50	466.948	317.260	733.479	549.900
50	48. 9559	20.7817	15.9336	35.8654	26.6549
100	5.25988	2.28902	1.83149	4.09419	3.06835
150	1.48466	0.650409	0.527554	1.17700	0.885485
200	0.611535	0.268725	0.219433	0.489114	0.368745
Parameters of interpolation Equation (8)
$d_0$, THz	4627.60	2017.90	1682.17	3741.20	2847.36
$d_1$	10.8572	12.5539	8.55139	8.95025	6.95394
$d_2$	270.400	89.4135	32.4966	48.3700	78.9251

for the μw dipole radiation transitions from the EIT-resonance-stimulating Rydberg $n{}_{}{}^1\mathrm{L}$ states with L = 0, 1, 2, 3 and some specific values of the principal quantum number in the region of $20\le n\le 200$ to $n^{\prime }{}_{}{}^1{\mathrm{L}}^{\prime }$ states of group IIb and Yb atoms. The notations used for the letters in subscripts of the quantities $∆E_{n\mathrm{L}{\mathrm{L}}^{\prime }}$ are as follows: the first letter n determines the principal quantum number of the $n{}_{}{}^1\mathrm{L}$ Rydberg state; the second letter L determines the angular momentum of the higher-energy state in the μw radiation transition; the third letter ${\mathrm{L}}^{\prime }$ determines the angular momentum of the state with lower energy.

The standard procedure of the curve fitting polynomial interpolation was used for the calculated data to derive analytical equations for the transition energy as functions of the $n{{}_{}{}^1\mathrm{L}}_{\mathrm{L}}$ state principal quantum number in the following form:

$$∆E_{n\mathrm{L}{\mathrm{L}}^{\prime }}=\frac{d_0}{n^3}\left(1+\frac{d_1}{n}+\frac{d_2}{n^2}\right).$$

(8)

The coefficients d₀, d₁, and d₂, presented in Table 2, Table 3, Table 4 and Table 5, were determined from specific numerical values of the transition energy $∆E_{n\mathrm{L}{\mathrm{L}}^{\prime }}$ for n = 20, 60, and 120. Evidently, the coefficients of d are closely related to coefficients ${\mu }_{2q}$ of resolution (5) for the quantum defects of states $n{}_{}{}^1\mathrm{L}$ and $n^{\prime }{}_{}{}^1{\mathrm{L}}^{\prime }$. Nevertheless, the straightforward use of the calculated data for transition frequencies appears more efficient for deriving the d coefficients of the asymptotic presentation (8). This conclusion follows from the fact that the quantum defects (7) are determined from energies (6) of only one and the same series of states with a fixed orbital quantum number L, whereas the transition energies (8) involve energies of two separate series of states with different orbital momenta L and L^′, and therefore with different quantum defects (7), involving implicitly terms of higher orders of the number $q>q_{\mathrm{m}\mathrm{a}\mathrm{x}}$.

4. Amplitudes of the μw Dipole Transitions between the Singlet Rydberg States of the Alkaline–Earth–Metal-like Atoms

The matrix element $\mathcal{R}=⟨n^{\prime }|z|n⟩$ of the single-electron dipole radiation transition between highly excited Rydberg states may be calculated using the standard methods of atomic spectroscopy [17,19]. Let the z-axis point along the polarization vectors of all linearly polarized radiation beams, i.e., the probe, coupling, and μw electromagnetic waves. Then, the z-axis may be considered as a quantization axis for the initial (ground) $n_0{{}_{ }{}^1\mathrm{S}}_0$ state, resonance $n_0{{}_{ }{}^1\mathrm{P}}_1,$ and Rydberg $n{{}_{}{}^1\mathrm{L}}_{\mathrm{L}}$ and $n^{\prime }{{}_{}{}^1{\mathrm{L}}^{\prime }}_{{\mathrm{L}}^{\prime }}$ states. Therefore, the magnetic quantum numbers of all involved states coincide with the z-component of the total angular momentum of the ground state $M_0=0$. After integration over angular variables with the use of the quantum theory of angular momentum [25], the μw transition matrix element may be presented in terms of the radial matrix element $⟨n^{\prime }{\mathrm{L}}^{\prime }|r|n\mathrm{L}⟩$, as follows:

$$\mathcal{R}=\frac{\mathrm{L}+{\mathrm{L}}^{\prime }+1}{2\sqrt{\left(2\mathrm{L}+1\right)\left(2{\mathrm{L}}^{\prime }+1\right)}}⟨n^{\prime }{\mathrm{L}}^{\prime }|r|n\mathrm{L}⟩.$$

(9)

The radial matrix element $⟨n^{\prime }{\mathrm{L}}^{\prime }|r|n\mathrm{L}⟩$ in this equation may be calculated using one of the well-known semi-empirical methods, the Fues’ Model Potential (FMP), or the Quantum Defect Method (QDM) [17,19]. In both methods, the radial wave functions may be presented in terms of polynomials in powers of their arguments

$$R_{n\mathrm{L}}^{\left(\mathrm{F}\mathrm{M}\mathrm{P}\right)}\left(r\right)=\frac{2Z^{3/2}}{{\nu }_{n\mathrm{L}}^2}\sqrt{\frac{n_r!}{\mathsf{\Gamma }\left({\nu }_{n\mathrm{L}}+\lambda +1\right)}}\exp \left(-\frac{x}{2}\right)x^{\lambda }{L}_{n_r}^{2\lambda +1}\left(x\right);$$

(10)

$$R_{n\mathrm{L}}^{\left(\mathrm{Q}\mathrm{D}\mathrm{M}\right)}\left(r\right)=\frac{2Z^{3/2}}{{\nu }_{n\mathrm{L}}^2\sqrt{\mathsf{\Gamma }\left({\nu }_{n\mathrm{L}}+\mathrm{L}+1\right)\mathsf{\Gamma }\left({\nu }_{n\mathrm{L}}-\mathrm{L}\right)}}\frac{W_{{\nu }_{n\mathrm{L}},\mathrm{L}+1/2}\left(x\right)}{x},$$

(11)

where $\mathsf{\Gamma }\left(z\right)$ is the gamma function,

$$L_{n_r}^{2\lambda +1}\left(x\right)=\frac{{\left(2\lambda +2\right)}_{n_r}}{n_r!}\sum _{k=0}^{n_r}\frac{{\left(-n_r\right)}_k}{k!{\left(2\lambda +2\right)}_k}{x}^k$$

(12)

is the generalized Laguerre polynomial [26], the argument $x=2Zr/{\nu }_{n\mathrm{L}}$ includes the Rydberg electron radial variable r, and the effective principal quantum number ${\nu }_{n\mathrm{L}}\equiv n-{\mu }_{n\mathrm{L}}=Z/\sqrt{-2E_{n\mathrm{L}}}$, related to the effective orbital $\lambda ={\nu }_{n\mathrm{L}}-n_r-1$ and radial $n_r=0,1,2,\dots$ quantum numbers [11,17], Z is the charge of residual ion (Z = 1 for a neutral atom), and ${\left(a\right)}_k=\mathsf{\Gamma }\left(a+k\right)/\mathsf{\Gamma }\left(a\right)=a·\left(a+1\right)·\dots ·\left(a+k-1\right)$ is the Pochhammer symbol [26]. The integer value $n_r$ determines the power of the Laguerre polynomial (12). Here, $E_{n\mathrm{L}}=-Z^2/\left(2{\nu }_{n\mathrm{L}}^2\right)$ is the Rydberg-state energy.

The Whittaker function of Equation (11) may be also presented in terms of a hypergeometric polynomial [26,27]

$${}_2{}^{ }{\tilde{F}}_0\left(a_1,a_2;-\frac{1}{x}\right)=\sum _{k=0}^{k_{\mathrm{m}\mathrm{a}\mathrm{x}}}\frac{{\left(a_1\right)}_k{\left(a_2\right)}_k}{k!}{\left(-\frac{1}{x}\right)}^k$$

(13)

in the form [25,26,27]:

$$W_{{\nu }_{n\mathrm{L}},\mathrm{L}+1/2}\left(x\right)=\exp \left(-\frac{x}{2}\right)x^{{\nu }_{n\mathrm{L}}}{}_2{}^{ }{\tilde{F}}_0\left(\mathrm{L}+1-{\nu }_{n\mathrm{L}},-{\nu }_{n\mathrm{L}};-\mathrm{L};-\frac{1}{x}\right).$$

(14)

The symbol tilde determines the polynomial, including a finite number of terms from an infinite number of terms in the hypergeometric series, corresponding to an asymptotic expansion of the Whittaker function in the vicinity of the origin r = 0. The maximal value of the summation index in (13) (the power of the polynomial) should be taken as $k_{\mathrm{m}\mathrm{a}\mathrm{x}}=\left[{\nu }_{n\mathrm{L}}\right]$ to eliminate the singularity of the radial wave function (11) [19]. The brackets $\left[a\right]$ indicate the integer part of $a$. Thus, the Whittaker function (14) remains finite at $x=0,$ despite the polynomial (13) singularity.

It is worth noting that the number of terms in the sum (12) $n_r$ may differ essentially from that of the sum (13) $k_{\mathrm{m}\mathrm{a}\mathrm{x}}$, since the integer part of the effective principal quantum number $\left[{\nu }_{n\mathrm{L}}\right]$ of states with a large orbital momentum L may exceed the radial quantum number $n_r$. This means that the terms with small powers of the radial variable in functions $R_{n\mathrm{L}}^{\left(\mathrm{F}\mathrm{M}\mathrm{P}\right)}\left(r\right)$ and $R_{n\mathrm{L}}^{\left(\mathrm{Q}\mathrm{D}\mathrm{M}\right)}\left(r\right)$ in (10) and (11) may be different. Meanwhile, the largest powers of the arguments are identical since $\lambda +n_r={\nu }_{n\mathrm{L}}-1$. Thus, the FMP and QDM functions differ from one another at small distances r, and they are practically identical at large distances from the atomic core. Therefore, the values of the matrix elements of dipole transitions between states with close energies, determined using FMP and QDM wave functions, practically coincide with each other. The agreement between the results of the FMP and QGM improves with an increase in the principal n and orbital L quantum numbers.

Using the wave functions (10) and (11), the integration of the radial matrix elements may be performed in the analytical form and presented in terms of combinations of the hypergeometric functions, such as [27]

$$\begin{gathered} {\tilde{\Sigma }}_4\left(a,b,a^{\prime },b^{\prime };c;z,z^{\prime }\right)=\sum _{k=0}^{k_{\mathrm{m}\mathrm{a}\mathrm{x}}}\sum _{k^{\prime }=0}^{{k^{\prime }}_{\mathrm{m}\mathrm{a}\mathrm{x}}}\frac{{\left(a\right)}_k{\left(b\right)}_k{\left(a^{\prime }\right)}_{k^{\prime }}{\left(b^{\prime }\right)}_{k^{\prime }}}{k!k^{\prime }!{\left(c\right)}_{k+k^{\prime }}}{z}^k{z^{\prime }}^{k^{\prime }} \\ =\sum _{k^{\prime }=0}^{k_{\mathrm{m}\mathrm{a}\mathrm{x}}}\frac{{\left(a\right)}_k{\left(b\right)}_k}{k!{\left(c\right)}_k}{z}^k{}_2{}^{ }{\tilde{F}}_1\left(a^{\prime },b^{\prime };c+k;z^{\prime }\right), \end{gathered}$$

(15)

where ${}_2{}^{ }{\tilde{F}}_1\left(a^{\prime },b^{\prime };c+k;z^{\prime }\right)$ is the Gauss hypergeometric function [26]. Finally, the analytic equation for the radial matrix element reads

$$⟨n^{\prime }{\mathrm{L}}^{\prime }|r|n\mathrm{L}⟩=\frac{x^{{\nu }^{\prime }+1}{\left(x^{\prime }\right)}^{\nu +1}\mathsf{\Gamma }\left(\nu +{\nu }^{\prime }+2\right){\tilde{\Sigma }}_4\left(a,b,a^{\prime },b^{\prime };c;1/x,1/x^{\prime }\right)}{4Z\sqrt{\mathsf{\Gamma }\left(\nu +\mathrm{L}+1\right)\mathsf{\Gamma }\left(\nu -\mathrm{L}\right)\mathsf{\Gamma }\left({\nu }^{\prime }+{\mathrm{L}}^{\prime }+1\right)\mathsf{\Gamma }\left({\nu }^{\prime }-{\mathrm{L}}^{\prime }\right)}},$$

(16)

where $x=2{\nu }^{\prime }/\left(\nu +{\nu }^{\prime }\right)$, $x^{\prime }=2\nu /\left(\nu +{\nu }^{\prime }\right)$, $a=\mathrm{L}+1-\nu$, $b=-\mathrm{L}-\nu$, $a^{\prime }={\mathrm{L}}^{\prime }+1-b=-\mathrm{L}-{\nu }^{\prime }$, $b^{\prime }=-{\mathrm{L}}^{\prime }-{\nu }^{\prime }$, $c=-1-\nu -{\nu }^{\prime }$. This equation was first presented in [27] with minor misprints, but when calculating numerical data, the correct Equation (16) was used. A similar equation for the radial matrix elements in terms of the hypergeometric polynomials of two variables may also be derived with the use of the FMP wave functions (see, for example, Section 4.5 of the reference [17]). The difference between the data of the QDM and FMP approaches does not exceed the uncertainties of the 1–3% characteristic of the semiempirical methods based on currently available data for atomic energy levels. The values of the parameters of the functions (10) and (11) are determined from the energy spectra of the ¹S₀, ¹P₁, ¹D_2, and ¹F₃ series of states of a given atom. For Rydberg states, the effective quantum numbers were calculated using the numerical values of the corresponding quantum defects, as presented in Section 3.

The results of the numerical computations with the use of Equations (15) and (16) for the matrix elements of the μw dipole transitions between Rydberg states with principal quantum numbers in the region from n = 10 to n = 250 are in good agreement with the existing data in the literature, with a fractional departure below 1%. The values of the matrix elements (9) may be conveniently approximated by a quadratic polynomial in the powers of the corresponding principal quantum number as follows:

$$\mathcal{R}\left(n\right)=a_0+a_{1}n+a_2{n}^2.$$

(17)

The coefficients $a_0$, $a_1$, and $a_2$, presented in

Table 6. Numerical values (in atomic units) and coefficients of the quadratic polynomial presentation (17) for the matrix elements of electrodipole transitions between the Rydberg states of Zn atoms. The bra-vector states are the states of higher energy, and the ket-vector states are the lower-energy states. Corresponding transition frequencies are presented in Table 2.

n	$⟨\mathit{n}{{}_{}{}^1\mathbf{S}}_0|\mathit{z}|\left(\mathit{n}-1\right){{}_{}{}^1\mathbf{P}}_1⟩$	$⟨\mathit{n}{{}_{}{}^1\mathbf{P}}_1|\mathit{z}|\mathit{n}{{}_{}{}^1\mathbf{S}}_0⟩$	$⟨\left(\mathit{n}+1\right){{}_{}{}^1\mathbf{P}}_1|\mathit{z}|\mathit{n}{{}_{}{}^1\mathbf{D}}_2⟩$	$⟨\mathit{n}{{}_{}{}^1\mathbf{D}}_2|\mathit{z}|\left(\mathit{n}-2\right){{}_{}{}^1\mathbf{F}}_3⟩$	$⟨\left(\mathit{n}-1\right){{}_{}{}^1\mathbf{F}}_3|\mathit{z}|\mathit{n}{{}_{}{}^1\mathbf{D}}_2⟩$
20	192.207	192.940	264.753	94.0656	262.173
50	1484.28	1378.05	1799.39	728.350	1742.94
100	6336.07	5756.48	7391.21	3102.77	7101.89
150	14,560.5	13,138.9	16,775.5	7123.17	16,075.02
200	26,157.7	23,525.3	29,952.4	12,789.5	28,662.3
Coefficients of interpolation polynomial (17)
$a_0$	5.19040	3.61539	0.0868455	−0.104969	−1.85717
$a_1$	−4.14509	−2.55116	−1.93900	−1.89060	−1.24583
$a_2$	0.674539	0.600798	0.758503	0.329194	0.722833

,

Table 7. Same, as in Table 6, for Cd atoms. Corresponding transition frequencies are presented in Table 3.

n	$⟨\mathit{n}{{}_{}{}^1\mathbf{S}}_0|\mathit{z}|\left(\mathit{n}-1\right){{}_{}{}^1\mathbf{P}}_1⟩$	$⟨\mathit{n}{{}_{}{}^1\mathbf{P}}_1|\mathit{z}|\mathit{n}{{}_{}{}^1\mathbf{S}}_0⟩$	$⟨\left(\mathit{n}+1\right){{}_{}{}^1\mathbf{P}}_1|\mathit{z}|\mathit{n}{{}_{}{}^1\mathbf{D}}_2⟩$	$⟨\mathit{n}{{}_{}{}^1\mathbf{D}}_2|\mathit{z}|\mathit{n}{{}_{}{}^1\mathbf{P}}_1⟩$	$⟨\left(\mathit{n}-2\right){{}_{}{}^1\mathbf{F}}_3|\mathit{z}|\mathit{n}{{}_{}{}^1\mathbf{D}}_2⟩$
20	180.176	160.779	233.531	100.911	236.081
50	1515.662	1209.88	1703.78	704.385	1684.28
100	6616.56	5150.60	7151.06	2918.56	7013.87
150	15,311.4	11,831.0	16,344.04	6643.98	15,988.4
200	27,600.1	21,251.2	29,282.7	11,880.7	28,607.9
Coefficients of interpolation polynomial (17)
$a_0$	8.67569	8.89962	2.19374	1.46048	−0.335623
$a_1$	−5.79943	−3.37771	−3.42512	−1.05397	−2.75730
$a_2$	0.718783	0.547947	0.749138	0.302249	0.728994

,

Table 8. Same, as in Table 6, for Hg atoms. Corresponding transition frequencies are presented in Table 4.

n	$⟨\mathit{n}{{}_{}{}^1\mathbf{S}}_0|\mathit{z}|\left(\mathit{n}-1\right){{}_{}{}^1\mathbf{P}}_1⟩$	$⟨\mathit{n}{{}_{}{}^1\mathbf{P}}_1|\mathit{z}|\mathit{n}{{}_{}{}^1\mathbf{S}}_0⟩$	$⟨\left(\mathit{n}+1\right){{}_{}{}^1\mathbf{P}}_1|\mathit{z}|\mathit{n}{{}_{}{}^1\mathbf{D}}_2⟩$	$⟨\mathit{n}{{}_{}{}^1\mathbf{D}}_2|\mathit{z}|\mathit{n}{{}_{}{}^1\mathbf{P}}_1⟩$	$⟨\left(\mathit{n}-2\right){{}_{}{}^1\mathbf{F}}_3|\mathit{z}|\mathit{n}{{}_{}{}^1\mathbf{D}}_2⟩$
20	162.316	137.992	218.701	64.7528	215.642
50	1450.13	1158.34	1698.30	445.609	1670.54
100	6464.05	5061.94	7258.25	1859.08	7127.96
150	15,061.8	11,718.3	16,685.6	4245.68	16,375.9
200	27,243.4	21,127.4	29,980.3	7605.33	29,414.4
Coefficients of interpolation polynomial (17)
$a_0$	20.0746	7.51134	5.75189	5.25138	3.66453
$a_1$	−7.23740	−4.51134	−4.82297	−0.924042	−4.56784
$a_2$	0.716772	0.550557	0.773479	0.194624	0.758108

and

Table 9. Same, as in Table 6, for Yb atoms. Corresponding transition frequencies are presented in Table 5.

n	$⟨\mathit{n}{{}_{}{}^1\mathbf{S}}_0|\mathit{z}|\left(\mathit{n}-1\right){{}_{}{}^1\mathbf{P}}_1⟩$	$⟨\mathit{n}{{}_{}{}^1\mathbf{P}}_1|\mathit{z}|\mathit{n}{{}_{}{}^1\mathbf{S}}_0⟩$	$⟨\mathit{n}{{}_{}{}^1\mathbf{D}}_2|\mathit{z}|\left(\mathit{n}+1\right){{}_{}{}^1\mathbf{P}}_1⟩$	$⟨\mathit{n}{{}_{}{}^1\mathbf{D}}_2|\mathit{z}|\left(\mathit{n}-2\right){{}_{}{}^1\mathbf{F}}_3⟩$	$⟨\left(\mathit{n}-1\right){{}_{}{}^1\mathbf{F}}_3|\mathit{z}|\mathit{n}{{}_{}{}^1\mathbf{D}}_2⟩$
20	101.472	201.632	216.582	136.886	193.186
50	943.043	1642.07	1610.04	1091.50	1398.82
100	4213.32	7143.57	6801.84	4721.86	5841.46
150	9818.61	16,519.6	15,580.7	10,901.4	13,324.6
200	17,758.9	29,770.0	27,946.6	19,630.0	23,848.1
Coefficients of interpolation polynomial (17)
$a_0$	7.79825	15.0819	5.30766	10.2918	−3.32539
$a_1$	−4.64536	−6.20523	−3.77607	−3.86720	−2.36219
$a_2$	0.467005	0.774901	0.717413	0.509829	0.608100

, are determined with the use of the standard curve fitting interpolation polynomial procedure for the calculated values of the matrix elements at n = 50, 100, and 150.

A comparison of the numerical values given by approximation (17) with matrix elements calculated in the FMP and QDM approaches confirms its high precision for all considered transitions in atoms; the fractional differences between calculated and approximated values do not exceed 0.1% in the regions of principal quantum numbers from 15 to 500.

5. Discussions

The interest in the highly excited states of the alkaline–earth–metal-like neutral Yb atom appeared in the 1980s, both in experimental and theoretical research [28,29,30]. In the 1990s, this interest reappeared and continued until the present day [22,31,32]. The interest in Rydberg-state Yb and alkaline–earth–metal-like atoms of group IIb elements is related, first of all, with the long-range interaction inducing practically important effects of Rydberg blockade [12] and electromagnetically induced transparency [33,34,35] of atomic gases, usually demonstrating strong opacity with respect to resonant light waves.

Due to the rather small amount of literature on the Rydberg-state energy levels of group IIb atoms, we had to use the databases of [13,14] for deriving quantum defects, presented in Table 1, for a series of singlet states with angular momenta $\mathrm{L}\le 3$. Thus, the data on transition frequencies between the closest ¹S₀, ¹P₁, ¹D_2, and ¹F₃ states were extended to Rydberg states with practically arbitrarily large principal quantum numbers n. The data in Table 2, Table 3, Table 4 and Table 5 and Equation (8) provide useful tools for determining the numerical values of the Rydberg–Rydberg dipole transition frequencies, which are absent in the current literature to date.

The high sensitivity of Rydberg states to static fields may influence the frequencies of the μw transitions calculated in Section 3 of this paper. Therefore, in measuring these frequencies, one should carefully reduce all residual, accidental, and stray fields of the laboratory apparatus, which could distort the measurement data. However, it is useful to bear in mind that the Stark shifts induced by intense high-frequency laser fields are nearly equal for all Rydberg states, conserving the immunity of the calculated transition frequencies in Section 2 to the field of the coupling waves.

Meanwhile, the intense coupling waves also influence the ground $n_0{{}_{ }{}^1\mathrm{S}}_0$ and resonance $n_0{{}_{ }{}^1\mathrm{P}}_1$ states, leading to the Stark shift of the absorbed probe radiation, thereby inducing the additional shifts of absorption and transparency for the probe wave. This effect should be taken into account together with the actions of environmental laboratory fields.

It is also worth noting that the data in Table 1, Table 2, Table 3, Table 4, Table 5, Table 6, Table 7, Table 8 and Table 9 may be useful for determining the most suitable atom and frequency of transition between its Rydberg states for evaluating the transition–amplitude-dependent efficiency in measuring the characteristics of the corresponding μw radiation. In this regard, the most important quantity that determines the transition amplitude is the coefficient a₂ of the quadratic polynomial (17). In particular, the maximal values of the transition amplitudes in the Zn, Cd, and Hg atoms correspond to (n + 1)¹P₁→n¹D₂ transitions. In Yb atoms, the maximal amplitudes correspond to the transitions n¹P₁→n¹S₀.

6. Conclusions

The main results of this paper are the numerical data for the basic characteristics of group IIb (Zn, Cd, Hg) and Yb atoms dipole radiation transitions from Rydberg state $n{{}_{}{}^1\mathrm{L}}_{\mathrm{L}}$ to the close energy states of opposite parity $n^{\prime }{{}_{}{}^1{\mathrm{L}}^{\prime }}_{{\mathrm{L}}^{\prime }}$ (where ${\mathrm{L}}^{\prime }=\mathrm{L}\pm 1$, $\left|n^{\prime }-n\right|\le 2$). The corresponding frequencies were determined for microwaves (μw), which are sometimes also called radio frequency (RF) or millimeter waves in the literature [23,24]. The most reliable data in the literature on energy levels were used to determine the quantum defects of n¹S₀, n¹P₁, n¹D_2, and n¹F₃ series of states (see Table 1) used for calculating the frequencies of transitions between highly excited Rydberg states. Equation (8) for the Rydberg–Rydberg transition frequencies as functions of the principal quantum number n was derived using the curve fitting interpolation polynomial approach with the coefficients shown in Table 2, Table 3, Table 4 and Table 5. This equation allows simple evaluations of frequencies for relevant electric dipole transitions between Rydberg states with close principal quantum numbers in the alkaline–earth–metal-like atoms.

The matrix elements of transitions $\mathcal{R}=⟨n^{\prime }|z|n⟩$ were calculated using the semi-empirical methods of Fues’ model potential (FMP) and the quantum defect method (QDM). The calculated numerical data demonstrated significant equivalence between the two methods; the fractional departure between the corresponding matrix elements did not exceed 0.1% in the region of the quantum number values n > 15. Therefore, only the results of the calculations in the QDM are presented in Table 6, Table 7, Table 8 and Table 9. The data for the matrix elements correspond to the transitions for which the data on frequencies may be found in Table 2, Table 3, Table 4 and Table 5. The curve fitting interpolation procedure, based on the numerical data for $\mathcal{R}\left(n\right)$ at n = 50, 100, and 150, was used for deriving the polynomials of the asymptotic presentation (17). The coefficients of the polynomials are listed for each transition in Table 6, Table 7, Table 8 and Table 9, thereby providing interpolated numerical values of matrix elements with a fractional departure from computed in the QDM approach data below 0.1% in the region of the principal quantum numbers n between 15 and 500.

In summary, the calculated results of this study provide new information on the frequencies and matrix elements of transitions between the highly excited Rydberg states of alkaline–earth–metal-like atoms, which so far are presented in the literature only for particular states of alkali and alkaline–earth–metal atoms [10,11]. The calculated numerical data may provide important information for planning further research on the use of Rydberg atoms for the development of new methods of μw radiation metrology and for constructing Rydberg-atom-based radio frequency systems for digital communications [36,37].

In addition to the effects on Rydberg singlet states of group IIb atoms and Yb, discussed in this paper, and those of group IIa, presented in [11], similar effects on triplet states may attract interest [38] and should be considered in future research.