Microwave Field Metrology Based on Rydberg States of Alkali-Metal Atoms

Abstract

The high-precision determination of microwave radiation parameters may be based on measurements of the spectral characteristics of radiation transitions between the Rydberg states of atoms. Frequencies and matrix elements are calculated for dipole transitions from even-parity nS_1/2 and nD_5/2 to odd-parity $n\prime$P_3/2 and $n\prime$F_7/2 (where $n\prime$ = n, n ± 1, n ± 2) for the Rydberg states of alkali-metal atoms. The matrix elements determine the splitting of Rydberg-state energy levels in the field of a resonance microwave (μw) radiation, which results in the splitting of the resonance in electromagnetic induced transparency (EIT). 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 of the current literature on quantum defect values were performed for the ²S, ²P, ²D and ²F series of the Rydberg states of Li, Na, K, Rb and Cs atoms. The calculated data were approximated by quadratic polynomials of the principal quantum number. The polynomial coefficients were determined with the use of a standard curve-fitting interpolation polynomial procedure for numerically presented functions. The approximation equations may be used for the accurate evaluation of the frequencies and matrix elements of μw transitions in wide ranges of the Rydberg-state quantum numbers n >> 1.

1. Introduction

The definition of physical quantities on the basis of the quantum properties of atoms seems to be the most reliable approach to complete the highest-precision set of metrological standards. Time-frequency standards, based on atoms in optical lattices, represent the most efficient examples of metrological standards with record fractional uncertainties (below 10⁻¹⁸), which are currently continuing to attract significant researchers’ attention [1].

The splitting of the resonance of the electromagnetically induced transparency (EIT) for the most intensely absorbed line of the principal series of alkali atoms in the field of microwave (μw) radiation deserves attention as a method of the high-precision determination of the μw electric fields and frequencies [2,3,4,5]. The frequency of the absorbed line is determined by the energy of the transition from the ground state $n_0{}^2{\mathrm{S}}_{1/2}$ to the first excited state $n_0{}^2{\mathrm{P}}_{3/2}$, where n₀ = 2, 3, 4, 5, and 6 are the principal quantum numbers of the Li, Na, K, Rb, Cs atoms, respectively. The EIT effect on the probe radiation with the frequency ${\omega }_{\mathrm{p}}=E_{n_0{\mathrm{P}}_{3/2}}-E_{n_0{\mathrm{S}}_{1/2}}$ in the alkali atomic vapor appears under the action of the sufficiently strong laser radiation of the frequency ${\omega }_{\mathrm{Laser}}\approx {\omega }_{\mathrm{c}}=E_{n{\mathrm{D}}_{5/2}({\mathrm{S}}_{1/2})}- E_{n_0{\mathrm{P}}_{3/2}}$, coupling the excited state $n_0{}^2{\mathrm{P}}_{3/2}$ to a highly excited Rydberg state $n{ }^2{\mathrm{D}}_{5/2}$ or $n{ }^2{\mathrm{S}}_{1/2}$ with a principal quantum number n >> 1. The resonance of the EIT effect appears when the coupling laser frequency coincides with the transition frequency exactly, i.e., ${\omega }_{\mathrm{Laser}}\equiv {\omega }_{\mathrm{c}}$.

If, together with the coupling laser field, μw radiation is applied with a frequency of ${\omega }_{\mathsf{\mu }\mathrm{w}}=E_n-E_{n\prime }-\epsilon$ close to the frequency of transition ($\left|\epsilon \right|<<{\omega }_{\mathsf{\mu }\mathrm{w}}$) from $n{ }^2{\mathrm{D}}_{5/2}$ ($n{ }^2{\mathrm{S}}_{1/2}$) to a Rydberg state $|n\prime \rangle \equiv |n\prime { }^2{\mathrm{F}}_{7/2}\rangle$ or $|n\prime \rangle \equiv |n\prime { }^2{\mathrm{P}}_{3/2}\rangle$ with $n\prime =n,\hspace{1em}n\pm 1,\hspace{1em}n\pm 2,$ then the state $n{ }^2{\mathrm{D}}_{5/2}$ ($n{ }^2{\mathrm{S}}_{1/2}$), due to the resonance Stark effect, transforms into two possible superpositions of Rydberg states. The energies of these superpositions are determined as the solutions of the secular equations for degenerate quasienergy states [6,7,8,9,10,11]:

$$E_n^{\pm }=E_{n{D}_{5/2}(S_{1/2})}^{}\pm \Delta E(\epsilon ,\Omega )$$

(1)

where (hereafter the atomic system of units is used $e=m=ħ=1$)

$$\Delta E(\epsilon ,\Omega )=\frac{1}{2}\sqrt{{\epsilon }^2+{\Omega }^2}$$

(2)

Here, $\Omega =Fℛ$ is the amplitude of the μw transition between Rydberg states $|n \rangle$ and $|n\prime \rangle$, which in the literature is called “Rabi frequency”; $F$ is the μw electric field; and $ℛ=\langle n\prime \left|z\right|n\rangle$ is the matrix element of the electric dipole moment z-component. The energy shift of the $n{ }^2{\mathrm{D}}_{5/2}$ ($n{ }^2{\mathrm{S}}_{1/2}$) state results in the splitting of the EIT resonance for the probe radiation $\Delta {\omega }_{\mathrm{p}}={\omega }_{\mathrm{p}}^{+}-{\omega }_{\mathrm{p}}^{-}=2\Delta E(\epsilon ,\Omega )$, which is now observed at the frequencies ${\omega }_{\mathrm{p}}^{\pm }={\omega }_{\mathrm{p}}^{}\pm \Delta E(\epsilon ,\Omega )$. Thus, the determination of the splitting, $\Delta {\omega }_{\mathrm{p}}={\omega }_{\mathrm{p}}^{+}-{\omega }_{\mathrm{p}}^{-}$, which coincides exactly with the EIT resonance splitting (2), enables the measurement of the electric field $F$ of the μw radiation. The number of Rydberg states in alkali-metal atoms is practically infinite, while the selection of probe ${\omega }_{\mathrm{p}}$ and coupling ${\omega }_{\mathrm{c}}$ laser frequencies providing the two-photon transition from the ground state to Rydberg $n{ }^2{\mathrm{D}}_{5/2}$ ($n{ }^2{\mathrm{S}}_{1/2}$) states is not difficult [12,13,14]. Therefore facilities always exist for choosing a set of four bound states (ground $n_0{}^2{\mathrm{S}}_{1/2}$, resonance $n_0{}^2{\mathrm{P}}_{3/2}$, and a pair of Rydberg states $n$ and $n\prime$) providing measuring field F of μw radiation on the basis of the EIT splitting $\Delta {\omega }_{\mathrm{p}}$. For an exact coincidence of the frequency ${\omega }_{\mathsf{\mu }\mathrm{w}}$ with the frequency of transition between Rydberg states (that is, for $\epsilon =0$) the splitting $\Delta {\omega }_{\mathrm{p}}$ coincides with the Rabi frequency. Then, the electric field of μw radiation with a frequency exactly equal to that of a transition between Rydberg states, ${\omega }_{\mathsf{\mu }\mathrm{w}}=E_n-E_{n\prime }$, is:

$$F=\Delta {\omega }_{\mathrm{p}}/ℛ$$

(3)

This relation holds for an atom at rest, i.e., in the atomic center-of-mass reference frame (cmrf). The laser frequencies, as seen by an atom in motion, account for the Doppler-effect shifts. The sum of the probe and coupling frequencies ${\omega }_{\mathrm{p}}$ and ${\omega }_{\mathrm{c}}$ of two laser waves, providing the two-photon excitation of a Rydberg state in a laboratory reference frame, will experience Doppler-effect transformation in an atomic cmrf, dependent on the atom’s thermal velocity projections on the probe and coupling laser beams. In order to minimize the Doppler effect on the μw-induced shift (2), the probe and coupling laser beams should propagate in opposite directions along one and the same line [12,13,14] (see Figure 1). In this case, the Doppler shift of the ${\omega }_{\mathrm{p}}^{+}$ frequency in the cmrf coincides with that of ${\omega }_{\mathrm{p}}^{-}$ and cancels out in their difference $\Delta {\omega }_{\mathrm{p}}={\omega }_{\mathrm{p}}^{+}-{\omega }_{\mathrm{p}}^{-}$. As such, for counter propagating probe and coupling waves, the difference $\Delta {\omega }_{\mathrm{p}}$ is Doppler-effect insensitive (remains equal to $2\Delta E(\epsilon ,\Omega )$, independent of the atomic thermal velocity).

Thus, given the frequency of μw radiation in the frequency ranges of sub-GHz, GHz up to a few THz, Rydberg state $n{ }^2{\mathrm{D}}_{5/2}$ ($n{ }^2{\mathrm{S}}_{1/2}$) should be determined with a close $n\prime { }^2{\mathrm{P}}_{3/2}$ ($n\prime { }^2{\mathrm{F}}_{7/2}$), providing the equality ${\omega }_{\mathsf{\mu }\mathrm{w}}=\left|E_n-E_{n\prime }\right|$. After the calculation of the matrix element $ℛ=\langle n\prime \left|z\right|n\rangle$, the μw electric field may be evaluated from Equation (3).

In this paper, the frequencies and corresponding matrix elements of the μw dipole transitions between Rydberg states of the first group elements (alkali-metal atoms Li, Na, K, Rb and Cs) are determined. The most reliable data on the energy spectra and quantum defects of the S- P-, D- and F-series of bound states were used in the evaluation of the frequencies of the μw transitions between Rydberg states. The numerical results of calculations and their extrapolations to states with extreme large-principal quantum numbers are discussed in Section 2. The values of the μw transition matrix elements $ℛ$ are calculated in the single-electron approximation with the use of the Fues’ model potential (FMP) and the quantum defect method (QDM). Numerical results and quadratic polynomial approximations for the evaluation of the amplitudes of transitions between Rydberg states are presented in Section 3.

2. Frequencies of μw Transitions from nD_5/2 and nS_1/2 Rydberg States of Alkali–metal Atoms

The existing present databases on the energies of atomic bound states may serve as a source for the determination of the frequencies of radiation transitions between the Rydberg states of atoms. Modern laser systems provide access to highly excited states with the use of methods of multiphoton transition spectroscopy. The states n²D_5/2 and n²S_1/2 seem most suitable for the observation of μw transitions in alkali-metal atoms, because in the close vicinity of their energies there exist states n’ ²P_3/2 and n’ ²F_7/2 with close values of principal quantum numbers $n\prime =n, n\pm 1, n\pm 2$, the frequencies the of transitions to which locate in Tera-, Giga- and Mega-Hertz diapasons.

In the databases [15,16], the numerical values of the energy levels for the nS-, nP-, nD- and nF-series are given only for a finite number of states with n ≤ n_max, where n_max depends essentially on the orbital momentum of a series presented in a concrete database for a concrete atom. In particular, for sodium and cesium atoms, the number n_max in the database [16] exceeds 1.5 to 2 times the number n_max of the base [15]. Meanwhile, the difference between the energy of the n’ ²P_3/2 (n’ ²F_7/2) and n²D_5/2 (n²S_1/2) levels for $\left|n\prime -n\right|\le 2$ vanishes rapidly (as 1/n³) with the increase of the principal quantum number. Therefore, the determination of the transition energy from tables of energy levels [15,16] becomes impossible already for n > 20, as the terms in the differences $\Delta E_{n\mathrm{FD}}=E_{n \prime \mathrm{F}}{}_{{}_{7/2}}-E_{n{\mathrm{D}}_{5/2}}$ and $\Delta E_{n\mathrm{D}\left(\mathrm{S}\right)\mathrm{P}}=E_{n\mathrm{D}\left(\mathrm{S}\right)}-E_{n\prime \mathrm{P}}$ differ only in the sixth or seventh digit already for n > 15. Thus, for the most precise determination of $\Delta E_{n\mathrm{DP}}$, $\Delta E_{n\mathrm{FD}}$ and $\Delta E_{n\mathrm{SP}}$, the data for quantum defects of the bound states should be used. The definition of the quantum defect ${\mu }_{n\mathrm{L}}$ is based on the Rydberg equation for the bound state with a principal quantum number n and the orbital quantum number L:

$$E_{n\mathrm{L}}=Ip-\frac{R{y}_A}{{(n-{\mu }_{n\mathrm{L}})}^2}$$

(4)

with $Ip$ being the energy of a single-electron ionization of an atom from its ground state, and $R{y}_A$ being the Rydberg constant, taking into account the finite mass of atom A.

The quantum defect ${\mu }_{nL}$ for a series of states with a fixed orbital momentum L is practically independent of the principal quantum number [17], and may be presented as a resolution of the form [18,19,20]

$${\mu }_{n\mathrm{L}}={\displaystyle \sum _{q=0}^{q_{\max }}\frac{{\mu }_{2q}}{{(n-{\mu }_0)}^{2q}}}$$

(5)

where ${\mu }_{2q}$ ($q=0,1,2,\dots ,q_{\max }$) are constant parameters for a series of states with a fixed angular momentum L. The doublet nL_j states of alkali atoms are split by the spin-orbit interaction into two fine-structure substates $|n\mathrm{L}j\rangle$ with different total orbital momenta $j=\mathrm{L}\pm 1/2$. In this paper, we consider only states with maximal values of total momentum $j=\mathrm{L}+1/2$ and maximal statistical weights, for which the angular parts of matrix elements are maximal, thus providing principal contributions into amplitudes of interaction with external fields. Sufficient numbers of theoretical and experimental works were performed on the determination of the numerical values of constants ${\mu }_{2q}$, $Ip$ and $R{y}_A$ [18,19,20,21,22,23,24], providing high-precision values of the bound-state energies (4) in alkali-metal atoms (A = Li, Na, K, Rb, Cs). In

Table 1. Numerical data for constants determining the quantum defects and energies of alkali-metal atoms’ Rydberg states from Equations (4) and (5). The source references are given in brackets.

Series nLj	${\mathit{\mu }}_{2\mathit{q}}$	Atom
		Li [18]	Na [19]	K [20]	Rb [21,22]	Cs [23,24]
nS1/2	${\mu }_0$	0.39951183	1.34797	2.18020826	3.1311807	4.0493527
	${\mu }_2$	0.0282456	0.060	0.134534	0.1787	0.238100
	${\mu }_4$	0.02082123	0.008	0.0952	0	0.24688
	${\mu }_6$	−09793152	0.005	0.00211	0	0.06785
	${\mu }_8$	0	0	0	0	0.1135
nP3/2	${\mu }_0$	0.04716876	0.854627	1.71087854	2.64142	3.5589599
	${\mu }_2$	−0.02398188	0.11055	0.23233	0.295	0.392469
	${\mu }_4$	0.01548488	0.0367	0.1961	0	−0.67431
	${\mu }_6$	−0.16065777	0.0881	0.3716	0	22.3531
	${\mu }_8$	0	0	0	0	−92.289
nD5/2	${\mu }_0$	0.00194211	0.0149013	0.27715665	1.3464622	2.4663091
	${\mu }_2$	−0.00376875	−0.42472	−1.02493	−0.594	0.014964
	${\mu }_4$	−0.01563348	0.006401	−0.640	0	−0.45828
	${\mu }_6$	0.10335313	0	10.0	0	−0.25489
	${\mu }_8$	0	0	0	0	−19.69
nF7/2	${\mu }_0$	0.00030862	0.001629	0.0094576	0.016411	0.03346
	${\mu }_2$	−0.00099057	−0.00685	−0.0446	−0.0784	−0.191
	${\mu }_4$	−0.00739661	0	0	0	0
RyA (cm−1)		109728.7	109734.715	109735.771	109736.605	109736.86
Ip (cm−1)		43487.1142	41449.451	35009.814	33690.81	31406.471

, the most reliable values of the constants from the cited papers are presented. As follows from the numerical values of constants ${\mu }_{2q}$, for Rydberg states with n > 15, the principal contribution into the sum (5) comes from the two terms q = 0 and 1. $q_{\max }=4$ only for the nP_3/2 and nD_5/2 series of Cs atoms; for these states of the other atoms, $q_{\max }=3$. For the nF_7/2 series only in Li atoms $q_{\max }=2$, while in other atoms $q_{\max }=1$, as in all series of Rb atoms, as presented in Table 1.

In

Table 2. Frequencies of electric dipole transitions in Li atoms: from state nP_3/2 to the nS_1/2 state, $\Delta E_{n\mathrm{PS}}=E_{n{\mathrm{P}}_{3/2}}-E_{n{\mathrm{S}}_{1/2}}$; from state nD_5/2 to the nP_3/2 state, $\Delta E_{n\mathrm{DP}}=E_{n{\mathrm{D}}_{5/2}}-E_{n{\mathrm{P}}_{3/2}}$; from the (n + 1)P_3/2 state to nD_5/2 cocтoяниe, $\Delta E_{n\mathrm{PD}}=E_{(n+1){\mathrm{P}}_{3/2}}-E_{n{\mathrm{D}}_{5/2}}$; and from the nF_7/2 state to the nD_5/2 state $\Delta E_{n\mathrm{FD}}=E_{n{\mathrm{F}}_{7/2}}-E_{n{\mathrm{D}}_{5/2}}$.

n	$$\mathbf{\Delta }{\mathit{E}}_{\mathit{n}\mathbf{PS}}, \mathbf{GHz}$$	$$\mathbf{\Delta }{\mathit{E}}_{\mathit{n}\mathbf{DP}}, \mathbf{GHz}$$	$$\mathbf{\Delta }{\mathit{E}}_{\mathit{n}\mathbf{PD}}, \mathbf{GHz}$$	$$\mathbf{\Delta }{\mathit{E}}_{\mathit{n}\mathbf{FD}}, \mathbf{GHz}$$
20	299.856	37.2897	732.592	1.33785
50	18.7973	2.38351	48.855	0.085923
100	2.33377	0.29776	6.1932	0.010746
150	0.689935	0.088206	1.8437	0.0031841
200	0.290740	0.037207	0.77964	0.0013433
Parameters of interpolation Equation (6)
d0, THz	2318.13	297.554	6281.52	10.7470
d1	0.669492	0.073775	−1.42344	0.00387
d2	0.537697	−0.44879	1.67349	−1.7229

,

Table 3. Frequencies of electric dipole transitions in Na atoms: from state nP_3/2 to the nS_1/2 state, $\Delta E_{n\mathrm{PS}}=E_{n{\mathrm{P}}_{3/2}}-E_{n{\mathrm{S}}_{1/2}}$; from state nD_5/2 to the nP_3/2 state, $\Delta E_{n\mathrm{DP}}=E_{n{\mathrm{D}}_{5/2}}-E_{n{\mathrm{P}}_{3/2}}$; from state (n + 1)P_3/2 to the nD_5/2 state, $\Delta E_{n\mathrm{PD}}=E_{(n+1){\mathrm{P}}_{3/2}}-E_{n{\mathrm{D}}_{5/2}}$; and from state nF_7/2 to the nD_5/2 state, $\Delta E_{n\mathrm{FD}}=E_{n{\mathrm{F}}_{7/2}}-E_{n{\mathrm{D}}_{5/2}}$.

n	$$\mathbf{\Delta }{\mathit{E}}_{\mathit{n}\mathbf{PS}}, \mathbf{GHz}$$	$$\mathbf{\Delta }{\mathit{E}}_{\mathit{n}\mathbf{DP}}, \mathbf{GHz}$$	$$\mathbf{\Delta }{\mathit{E}}_{\mathit{n}\mathbf{PD}}, \mathbf{GHz}$$	$$\mathbf{\Delta }{\mathit{E}}_{\mathit{n}\mathbf{FD}}, \mathbf{GHz}$$
20	480.9481	738.746	130.232	10.8558
50	27.7624	45.3832	8.40018	0.698217
100	3.35562	5.59802	1.05227	0.087240
150	0.983267	1.65126	0.312084	0.0260819
200	0.412523	0.694919	0.131909	0.0110923
Parameters of interpolation Equation (6)
d0, THz	3246.32	5524.72	1054.30	87.8673
d1	3.28154	1.30589	−0.176685	−0.475097
d2	8.45437	1.77471	−1.18746	4.85415

,

Table 4. Frequencies of electric dipole transitions in K atoms: from state nP_3/2 to the nS_1/2 state, $\Delta E_{n\mathrm{PS}}=E_{n{\mathrm{P}}_{3/2}}-E_{n{\mathrm{S}}_{1/2}}$; from state nD_5/2 to the (n + 1)P_3/2 state, $\Delta E_{n\mathrm{DP}}=E_{n{\mathrm{D}}_{5/2}}-E_{n{\mathrm{P}}_{3/2}}$; from state (n + 2)P_3/2 to the nD_5/2 state, $\Delta E_{n\mathrm{PD}}=E_{(n+2){\mathrm{P}}_{3/2}}-E_{n{\mathrm{D}}_{5/2}}$; and from state nF_7/2 to the nD_5/2 state, $\Delta E_{n\mathrm{FD}}=E_{n{\mathrm{F}}_{7/2}}-E_{n{\mathrm{D}}_{5/2}}$.

n	$$\mathbf{\Delta }{\mathit{E}}_{\mathit{n}\mathbf{PS}}, \mathbf{GHz}$$	$$\mathbf{\Delta }{\mathit{E}}_{\mathit{n}\mathbf{DP}}, \mathbf{GHz}$$	$$\mathbf{\Delta }{\mathit{E}}_{\mathit{n}\mathbf{PD}}, \mathbf{GHz}$$	$$\mathbf{\Delta }{\mathit{E}}_{\mathit{n}\mathbf{FD}}, \mathbf{GHz}$$
20	524.637	387.441	462.795	222.820
50	27.8267	23.5484	29.7713	14.1916
100	3.27545	2.89719	3.72462	1.76830
150	0.951504	0.85381	1.10384	0.523295
200	0.397486	0.35945	0.46558	0.220623
Parameters of interpolation Equation (6)
d0, THz	3090.00	2853.86	3725.82	1761.28
d1	5.70541	1.47176	−0.01489	0.437875
d2	29.2063	4.99768	−2.22060	−3.92491

,

Table 5. Frequencies of electric dipole transitions in Rb atoms: from state nP_3/2 to the nS_1/2 state, $\Delta E_{n\mathrm{PS}}=E_{n{\mathrm{P}}_{3/2}}-E_{n{\mathrm{S}}_{1/2}}$; from state nD_5/2 to the (n + 1)P_3/2 state, $\Delta E_{n\mathrm{DP}}=E_{n{\mathrm{D}}_{5/2}}-E_{(n+1){\mathrm{P}}_{3/2}}$; from state (n + 2)P_3/2 to the nD_5/2 state, $\Delta E_{n\mathrm{PD}}=E_{(n+2){\mathrm{P}}_{3/2}}-E_{n{\mathrm{D}}_{5/2}}$; and from state (n−1)F_7/2 to the nD_5/2 state, $\Delta E_{n\mathrm{FD}}=E_{(n-1){\mathrm{F}}_{7/2}}-E_{n{\mathrm{D}}_{5/2}}$.

n	$$\mathbf{\Delta }{\mathit{E}}_{\mathit{n}\mathbf{PS}}, \mathbf{GHz}$$	$$\mathbf{\Delta }{\mathit{E}}_{\mathit{n}\mathbf{DP}}, \mathbf{GHz}$$	$$\mathbf{\Delta }{\mathit{E}}_{\mathit{n}\mathbf{PD}}, \mathbf{GHz}$$	$$\mathbf{\Delta }{\mathit{E}}_{\mathit{n}\mathbf{FD}}, \mathbf{GHz}$$
20	642.809	308.911	673.701	324.384
50	30.8125	17.0269	39.39871	18.6529
100	3.51836	2.03100	4.77955	2. 25009
150	1.01210	0.592638	1.40212	0.658844
200	0.420760	0.248127	0.588582	0.276310
Parameters of interpolation Equation (6)
d0, THz	3230.09	1941.67	4639.03	2171.68
d 1	8.17548	4.38292	2.97713	3.53804
d 2	73.3112	21.4566	5.17588	7.22242

and

Table 6. Frequencies of electric dipole transitions in Cs atoms: from state nP_3/2 to the nS_1/2 state, $\Delta E_{n\mathrm{PS}}=E_{n{\mathrm{P}}_{3/2}}-E_{n{\mathrm{S}}_{1/2}}$; from state nD_5/2 to the (n + 1)P_3/2 state, $\Delta E_{n\mathrm{DP}}=E_{n{\mathrm{D}}_{5/2}}-E_{(n+1){\mathrm{P}}_{3/2}}$; from state (n + 2)P_3/2 to state nD_5/2, $\Delta E_{n\mathrm{PD}}=E_{(n+2){\mathrm{P}}_{3/2}}-E_{n{\mathrm{D}}_{5/2}}$; from state (n−2)F_7/2 to state nD_5/2, $\Delta E_{n\mathrm{FD}}=E_{(n-2){\mathrm{F}}_{7/2}}-E_{n{\mathrm{D}}_{5/2}}$; and from state nS_1/2 to state (n−1)P_3/2, $\Delta E_{n\mathrm{SP}}=E_{n{\mathrm{S}}_{1/2}}-E_{(n-1){\mathrm{P}}_{3/2}}$.

n	$$\mathbf{\Delta }{\mathit{E}}_{\mathit{n}\mathbf{PS}}, \mathbf{GHz}$$	$$\mathbf{\Delta }{\mathit{E}}_{\mathit{n}\mathbf{DP}}, \mathbf{GHz}$$	$$\mathbf{\Delta }{\mathit{E}}_{\mathit{n}\mathbf{PD}}, \mathbf{GHz}$$	$$\mathbf{\Delta }{\mathit{E}}_{\mathit{n}\mathbf{FD}}, \mathbf{GHz}$$	$$\mathbf{\Delta }{\mathit{E}}_{\mathit{n}\mathbf{SP}}, \mathbf{GHz}$$
20	759.2494	115.5336	1025.984	510.1322	868.9888
50	32.72700	5.70305	54.0257	26.16514	35.14842
100	3.624683	0.658255	6.34553	3.049394	3.826316
150	1.0326152	0.190051	1.84206	0.883014	1.084192
200	0.4272438	0.079155	0.769246	0.368295	0.447403
Parameters of interpolation Equation (6)
d0, THz	3246.36	610.923	5974.424	2851.130	3385.774
d1	10.1607	7.10204	5.88922	6.52633	10.9155
d2	145.193	63.1217	31.7494	42.0264	202.998

, the numerical values of the frequencies are presented for the dipole radiation transitions from Rydberg n²S_1/2 and n²D_5/2 states, with some concrete values of the principal quantum number in the region 20 ≤ n ≤ 200, to the n’ ²P_3/2 and to n’ ²F_7/2 states of the alkali-metal atoms. The notations used for the letters in the subscripts of the quantities $\Delta E_{n\mathrm{LL}\prime }$ are as follows: the first letter n determines the principal quantum numbers of the two-photon accessible Rydberg states n²S_1/2 or n²D_5/2, the second letter L determines the angular momentum of higher-energy Rydberg states, and the third letter $\mathrm{L}\prime$ determines the angular momentum of lower-energy Rydberg states. The total orbital momentum for all of the states is assumed to have the maximal value of the doublet-state total momentum, j = L + 1/2.

The standard procedure of curve-fitting polynomial interpolation was used for the calculated data to derive analytical equations for the energy of transition as functions of the n²D_5/2- (n²S_1/2-)state principal quantum number in the form

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

(6)

The coefficients d₀, d₁, and d₂ of the quadratic polynomial, presented in Table 2, Table 3, Table 4, Table 5 and Table 6, were determined from concrete numerical values of the transition energy $\Delta E_{n\mathrm{LL}\prime }$ for n = 20, 60, 120.

Equation (6) appears to be rather convenient for simplified evaluations of the transition energy for arbitrary values of n, providing quite satisfactory precision. As such, the fractional departure of the values (6) from those given by Equations (4) and (5) does not exceed 0.1% in the region of n values between 10 and 500. In particular, for frequencies of transitions from n²D_5/2 to (n + 1)²P_3/2 states of rubidium atoms, Equation (6) with parameters from Table 5 reproduces the data of papers [2,13,14] with fractional departures below 1%. Furthermore, the frequency of transition 49²F_7/2→50²D_5/2, as presented in [2] with four digits, ${\omega }_{\mu w}=18.65 \Gamma \Gamma Ц$, coincides exactly with the result given by Equations (4) and (5) (see Table 5). This value is also reproduced by Equation (6) with parameters from Table 5. However, we should draw attention to Figure 8 of reference [2] with an error in the determination of the correspondence of the numerical values of frequencies 5.10 GHz and 10.22 GHz to transitions 74²D_5/2→75²P_3/2 and 59 ²D_5/2→60²P_3/2. Furthermore, it is necessary to note the agreement of results obtained from Equations (4)–(6) for the μw transitions between the Rydberg states of Cs atoms (see Table 6) with all of the numerical data of the papers [9,25,26].

3. Amplitudes of the Dipole Transitions from Rydberg nD_5/2 and nS_1/2 States to States n’P_3/2 and n’F_7/2

The matrix element $ℛ\left(n\right)=\langle n\prime \left|z\right|n\rangle$ of the single-electron dipole radiation transition between highly excited Rydberg states may be calculated with the use of standard methods of atomic spectroscopy [11,17]. Let the z-axis point along the polarization vectors of the three linear polarized radiations: the probe, coupling and μw beams. Then the z-axis may be considered as a quantization axis for the initial (ground) $n_0{}^2{\mathrm{S}}_{1/2}$, resonance $n_0{}^2{\mathrm{P}}_{3/2}$ and Rydberg ($n{ }^2{\mathrm{D}}_{5/2}$, $n{ }^2{\mathrm{S}}_{1/2}$, $n\prime {\mathrm{P}}_{3/2}$ and $n\prime {\mathrm{F}}_{7/2}$) states. The magnetic quantum numbers in these states coincide with the z-component of the total angular momentum of the ground state m = ±1/2. After integration over angular variables with the use of the quantum theory of angular momentum [27], the μw transition matrix element may be presented in terms of the radial matrix element, as follows:

(1) For transitions from nD_5/2 states to states $n\prime$P_3/2:

$$ℛ=\langle n\prime {\mathrm{P}}_{3/2}|z|n{\mathrm{D}}_{5/2}\rangle =\frac{\sqrt{6}}{5}\langle R_{n\prime {\mathrm{P}}_{3/2}}(r)|r|R_{n{\mathrm{D}}_{5/2}}(r)\rangle$$

(7)

(2) For transitions from nD_5/2 states to states $n\prime$F_7/2:

$$ℛ=\langle n\prime {\mathrm{F}}_{7/2}|z|n{\mathrm{D}}_{5/2}\rangle =\frac{2\sqrt{3}}{7}\langle R_{n\prime {\mathrm{F}}_{7/2}}(r)|r|R_{n{\mathrm{D}}_{5/2}}(r)\rangle$$

(8)

(3) For transitions from nS_1/2 states to states $n\prime$P_3/2:

$$ℛ=\langle n\prime {\mathrm{P}}_{3/2}|z|n{\mathrm{S}}_{1/2}\rangle =\frac{\sqrt{2}}{3}\langle R_{n\prime {\mathrm{P}}_{3/2}}(r)|r|R_{n{\mathrm{S}}_{1/2}}(r)\rangle$$

(9)

The calculation of the radial matrix elements $\langle R_{n\prime {\mathrm{P}}_{3/2}}(r)|r|R_{n{\mathrm{D}}_{5/2}}(r)\rangle$, $\langle R_{n\prime {\mathrm{F}}_{7/2}}(r)|r|R_{n{\mathrm{D}}_{5/2}}(r)\rangle$ and $\langle R_{n\prime {\mathrm{P}}_{3/2}}(r)|r|R_{n{\mathrm{S}}_{1/2}}(r)\rangle$ in these equations may be performed with the use of one of the well-known semiempirical methods, the Fues’ Model Potential (FMP) or the Quantum Defect Method (QDM) [11,17]. In both methods, the radial wave functions are presented in terms of polynomials in powers of arguments

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

(10)

$$R_{n\mathrm{L}}^{(\mathrm{QDM})}(r)=\frac{2Z^{3/2}}{{\nu }_{n\mathrm{L}}^2\sqrt{\Gamma ({\nu }_{n\mathrm{L}}+\mathrm{L}+1)\Gamma ({\nu }_{n\mathrm{L}}-\lambda )}} \frac{W_{{\nu }_{n\mathrm{L}}, \mathrm{L}+1/2}(x)}{x}$$

(11)

where $\Gamma \left(z\right)$ is the gamma function;

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

(12)

is the generalized Laguerre polynomial [28]; the argument $x=2Zr/{\nu }_{n\mathrm{L}}$ includes the radial variable $r$ and the effective principal quantum number ${\nu }_{n\mathrm{L}}\equiv n-{\mu }_{nL}=Z/\sqrt{-2E_{n\mathrm{L}}}$, which are related with the effective orbital $\lambda ={\nu }_{n\mathrm{L}}-n_r-1$ and radial $n_r=0,1,2,\dots$ quantum numbers [11]; Z is the charge of the residual ion (Z = 1 for a neutral atom); and ${(a)}_k=\Gamma (a+k)/\Gamma (a)=a\cdot (a+1)\cdot \dots \cdot (a+k-1)$ is the Pochhammer symbol [28]. The integer value $n_r$ determines the power of the Laguerre polynomial (12). Here, $E_{nL}=-Z^2/(2{\nu }_{nL}^2)$ is the Rydberg-state energy.

The Whittaker function of Equation (11) may also be presented in terms of a hypergeometric polynomial [28,29]

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

(13)

in the form [28,29,30]

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

(14)

The maximal value of the summation index (the power of polynomial) in (13) is k_max = [${\nu }_{n\mathrm{L}}$], where the brackets [a] determine the integer part of the value a. As such, the Whittaker function remains finite for x → 0 despite the singularity of the polynomial (13).

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

With the use of the wave functions (10) and (11), the integration in the radial matrix elements may be performed in analytical form. As such, the matrix elements are presented in terms of combinations of the hypergeometric functions, which may be evaluated numerically to a necessary precision with the use of currently available computation facilities. The values of the parameters of functions (10) and (11) are determined from the energy spectra of the S_1/2, P_3/2, D_5/2 and F_7/2 series of states of a given atom. For Rydberg states, the calculation of effective quantum numbers may be performed with the use of the numerical values of corresponding quantum defects, as was presented in Section 2 of this paper.

The results of numerical computations of the matrix elements (7–9) of the μw dipole transitions between Rydberg states with principal quantum numbers in the region from n = 10 to 250 give a good agreement with the data presented in the literature [2,8,9,13,14,25,26], with the fractional departure being below 1%. The values of the matrix elements, calculated in Section 2 for the μw transitions in each alkali-metal atom, may be conveniently presented in terms of a quadratic polynomial in powers of the corresponding principal quantum number, as follows:

$$ℛ(n)=a_0+a_{1}n+a_2{n}^2$$

(15)

The coefficients a₀, a₁, and a₂, presented in

Table 7. Numerical values (in atomic units) and coefficients of the quadratic polynomial presentation (15) for the matrix elements of electrodipole transitions from Rydberg nS_1/2 and nD_5/2 states of Li atoms. The bra-vector states are the states of higher energy, and the ket-vector states are the lower-energy states. The corresponding transition frequencies are presented in Table 2.

n	$\langle \mathit{n}{\mathbf{P}}_{3/2}|\mathit{z}|\mathit{n}{\mathbf{S}}_{1/2}\rangle$	$\langle \mathit{n}{\mathbf{D}}_{5/2}|\mathit{z}|\mathit{n}{\mathbf{P}}_{3/2}\rangle$	$\langle (\mathit{n}+1){\mathbf{P}}_{3/2}|\mathit{z}|\mathit{n}{\mathbf{D}}_{5/2}\rangle$	$\langle \mathit{n}{\mathbf{F}}_{7/2}|\mathit{z}|\mathit{n}{\mathbf{D}}_{5/2}\rangle$
20	254.05	295.13	71.160	293.59
50	1596.6	1850.9	465.70	1852.5
100	6396.6	7406.0	1890.6	7419.8
150	14,399.6	16,663.8	4274.5	16,698.7
200	25,605.7	29,624.4	7617.6	29,689.1
Coefficients of interpolation polynomial (15)
a0	−0.36230	−1.42531	−0.044123	−3.3464
a1	−0.09145	0.01966	−0.276371	0.001437
a2	0.64061	0.74055	0.1918234	0.742305

,

Table 8. The same as in Table 7, for Na atoms. The corresponding transition frequencies are presented in Table 3.

n	$\langle \mathit{n}{\mathbf{P}}_{3/2}|\mathit{z}|\mathit{n}{\mathbf{S}}_{1/2}\rangle$	$\langle \mathit{n}{\mathbf{D}}_{5/2}|\mathit{z}|\mathit{n}{\mathbf{P}}_{3/2}\rangle$	$\langle (\mathit{n}+1){\mathbf{P}}_{3/2}|\mathit{z}|\mathit{n}{\mathbf{D}}_{5/2}\rangle$	$\langle \mathit{n}{\mathbf{F}}_{7/2}|\mathit{z}|\mathit{n}{\mathbf{D}}_{5/2}\rangle$
20	191.977	116.823	281.937	293.749
50	1268.55	712.008	1776.68	1852.67
100	5166.38	2822.30	7120.68	7419.45
150	11,693.6	6330.49	16,030.6	16,697.02
200	20,850.3	11236.6	28,506.6	29,685.4
Coefficients of interpolation polynomial (15)
a0	0.15071	−0.378542	−1.35956	−3.34483
a1	−0.926165	0.268697	−0.098926	0.0167251
a2	0.525885	0.279581	0.713193	0.742156

,

Table 9. The same as in Table 7, for K atoms. The corresponding transition frequencies are presented in Table 4.

n	$\langle \mathit{n}{\mathbf{P}}_{3/2}|\mathit{z}|\mathit{n}{\mathbf{S}}_{1/2}\rangle$	$\langle \mathit{n}{\mathbf{D}}_{5/2}|\mathit{z}|(\mathit{n}+1){\mathbf{P}}_{3/2}\rangle$	$\langle (\mathit{n}+2){\mathbf{P}}_{3/2}|\mathit{z}|\mathit{n}{\mathbf{D}}_{5/2}\rangle$	$\langle \mathit{n}{\mathbf{F}}_{7/2}|\mathit{z}|\mathit{n}{\mathbf{D}}_{5/2}\rangle$
20	189.372	229.992	187.764	275.777
50	1324.10	1456.77	1210.18	1716.92
1000	5489.55	5845.80	4890.78	6848.27
150	12,497.8	13165.3	11,042.1	15,391.5
200	22,348.8	23,415.2	19,664.1	27,346.6
Coefficients of interpolation polynomial (15)
a0	1.45807	−1.81865	0.259006	−2.54978
a1	−1.97512	−0.132718	−0.508553	0.270670
a2	0.568560	0.586089	0.494138	0.682375

,

Table 10. The same as in Table 7, for Rb atoms. The corresponding transition frequencies are presented in Table 5.

n	$\langle \mathit{n}{\mathbf{P}}_{3/2}|\mathit{z}|\mathit{n}{\mathbf{S}}_{1/2}\rangle$	$\langle \mathit{n}{\mathbf{D}}_{5/2}|\mathit{z}|(\mathit{n}+1){\mathbf{P}}_{3/2}\rangle$	$\langle (\mathit{n}+2){\mathbf{P}}_{3/2}|\mathit{z}|\mathit{n}{\mathbf{D}}_{5/2}\rangle$	$\langle (\mathit{n}-1){\mathbf{F}}_{7/2}|\mathit{z}|\mathit{n}{\mathbf{D}}_{5/2}\rangle$
20	158.462	231.722	129.809	239.100
50	1183.28	1575.09	902.669	1579.85
1000	5004.32	6467.10	3737.47	6424.52
150	11,466.7	14,675.4	8505.65	14,531.9
200	20,570.3	26,200.0	15,207.2	25,901.9
Coefficients of interpolation polynomial (15)
a0	3.53096	−0.641255	1.23842	−2.15111
a1	−2.81802	−1.64817	−1.30512	−0.986724
a2	0.528259	0.663256	0.3866746	0.652534

and

Table 11. The same as in Table 7, for Cs atoms. The corresponding transition frequencies are presented in Table 6. The subscripts determining the total angular momentum j = L + ½ are omitted.

n	$\langle \mathit{n}\mathbf{P}|\mathit{z}|\mathit{n}\mathbf{S}\rangle$	$\langle \mathit{n}\mathbf{D}|\mathit{z}|(\mathit{n}+1)\mathbf{P}\rangle$	$\langle (\mathit{n}+2)\mathbf{P}|\mathit{z}|\mathit{n}\mathbf{D}\rangle$	$\langle (\mathit{n}-2)\mathbf{F}|\mathit{z}|\mathit{n}\mathbf{D}\rangle$	$\langle \mathit{n}\mathbf{S}|\mathit{z}|(\mathit{n}-1)\mathbf{P}\rangle$
20	142.005	222.961	65.13397	194.399	122.769
50	1136.96	1644.53	500.445	1367.25	1064.16
1000	4906.55	6924.06	2134.98	5668.55	4693.02
150	11,315.3	15,841.4	4905.83	12,903.1	10,896.8
200	20,363.3	28,396.6	8812.98	23,070.8	19,675.4
Coefficients of interpolation polynomial (15)
a0	6.57465	2.81058	2.21211	−0.854001	10.1763
a1	−3.78433	−3.54369	−1.39838	−1.99011	−4.66898
a2	0.527840	0.727562	0.227261	0.592597	0.514975

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

4. Discussion

The results of calculations of the dipole-transition matrix elements $ℛ=\langle n\prime \left|z\right|n\rangle$ are presented in Table 7, Table 8, Table 9, Table 10 and Table 11, although they were performed for components with only maximal total orbital momenta j = L + ½ of the doublet Rydberg states nLj of alkali-metal atoms, and may also be used for states with j = L − ½. The difference will arise in only the angular parts of $ℛ$, which are easily evaluated numerically on the basis of standard methods of the theory of angular momenta [27]. Meanwhile, the fractional difference of corresponding radial parts $\langle n\prime |r|n\rangle$ does not exceed 0.1% at n = 20, and this difference will decrease with an increasing n, following the decrease of the fine-structure intervals, ${\Delta }_{fs}(n,L)=E_{nLj=L+1/2}-E_{nLj=L-1/2}={\delta }_L/n^3$, where ${\delta }_L$ is a constant factor for a given series of doublet states nLj [17].

The compilation of the data on quantum defects, taken from the literature and presented in Table 1, provided possibilities for the determination of the frequencies of the μw transitions between Rydberg states. The results of these calculations, presented in Table 2, Table 3, Table 4, Table 5 and Table 6, together with coefficients of approximation polynomial (6), reproduce all the of the data of the literature with fractional uncertainty below 1%, thus confirming their reliability. The approximations (6) and (15) open new possibilities for the further extension of the numerical data on the energy levels and amplitudes of radiation transitions between Rydberg states of alkali-metal atoms.

The high sensitivity of Rydberg states to static fields may influence the frequencies of the μw transitions calculated in Section 2 of this paper. Therefore, in measuring these frequencies, one should take care of reducing all stray laboratory fields. 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 transition frequencies, calculated in Section 2, to the field of the coupling wave.

It is also worth noting, that the data of Table 2, Table 3, Table 4, Table 5, Table 6, Table 7, Table 8, Table 9, Table 10 and Table 11 may be useful for the determination of the most suitable atom and frequency of transition between its Rydberg states for the evaluation of the transition-amplitude-dependent efficiency of the measurement characteristics of corresponding μw radiation. In this regard, the most important quantity determining the transition amplitude is the coefficient a ₂ of the quadratic polynomial (15). In particular, the maximal values of the transition amplitudes in Li, Na and K atoms correspond to nF_7/2→nD_5/2 transitions. In Rb and Cs atoms, the maximal amplitudes correspond to transitions nD_5/2→(n + 1)P_3/2.

5. Conclusions

The main results of this paper are the numerical data for the basic characteristics of the dipole radiation transitions from Rydberg $n{ }^2{\mathrm{D}}_{5/2}$ and/or the $n{ }^2{\mathrm{S}}_{1/2}$ state to close in on the energy states of the opposite parity, $n\prime { }^2{\mathrm{P}}_{3/2}$ or $n\prime { }^2{\mathrm{F}}_{7/2}$, in the alkali-metal atoms Li, Na, K, Rb, Cs. The corresponding frequencies were determined, located in the microwave (μw) frequency region, which is sometimes called the radio frequency (rf) range in the literature. The most reliable data of the literature on the quantum defects of energy levels were used (see Table 1) for the calculation of the frequencies of transitions between highly excited Rydberg states. Equation (6) for the Rydberg–Rydberg transition frequencies as functions of the principal quantum number n was derived within the curve-fitting polynomial interpolation approach. Accompanied by the lists of parameters presented in Table 2, Table 3, Table 4, Table 5 and Table 6, this equation enables simple evaluations of frequencies for all types of electric dipole transitions between the Rydberg states of close-principal quantum numbers in alkali-metal atoms.

The matrix elements of transitions $ℛ=\langle n\prime \left|z\right|n\rangle$ were calculated with the use of the semiempirical methods of Fues’ model potential (FMP) and the quantum defect method (QDM). The calculated numerical data demonstrated the significant equivalence of the two methods; the fractional departure between corresponding numerical data did not exceed 0.1%. Therefore, in Table 7, Table 8, Table 9, Table 10 and Table 11 are presented only the results of calculations in the QDM. The data for matrix elements corresponds to the transitions for which the data on frequencies are presented in Table 2, Table 3, Table 4, Table 5 and Table 6. The curve-fitting polynomial interpolation procedure—based on numerical data for $ℛ(n)$ at n = 50, 100 and 150—was used to derive the polynomial presentation (15). The coefficients of the polynomial are listed for each transition in Table 7, Table 8, Table 9, Table 10 and Table 11, thereby providing interpolated numerical values of matrix elements with a fractional departure from the computed data below 0.1% in the region of the principal quantum numbers n between 15 and 500.

In summary, the calculated results of this article provide new information on the frequencies and matrix elements of transitions between the highly excited Rydberg states of alkali-metal atoms, which so far have been presented in the literature only for Rb and Cs atoms’ particular states, with some fixed principal quantum numbers. The calculated numerical data may provide important information for the planning of further research on the use of Rydberg atoms for the development of new methods of μw radiation metrology, and for the construction of Rydberg-atom-based radio frequency systems for digital communications [31].

Together with alkali-metal atoms, the Rydberg states of alkaline-earth-metal atoms, such as strontium, may also be useful for the determination of the characteristics of μw radiation [32]. To this end, the methods of the present paper may be used for the calculation of the data on the frequencies and amplitudes of μw transitions between the Rydberg states of the alkaline-earth atoms.