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

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 nS1/2 and nD5/2 to odd-parity n′P3/2 and n′F7/2 (where n′ = 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

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

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 nS1/2 and nD5/2 to odd-parity n′P3/2 and n′F7/2 (where n′ = 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

Van-der-Waals interaction of atoms in dipolar Rydberg states

An asymptotic expression for the van-der-Waals constant C6(n) ≈ –0.03n12Kp(x) is derived for the long-range interaction between two highly excited hydrogen atoms A and B in their extreme Stark states of equal principal quantum numbers nA = nB = n ≫ 1 and parabolic quantum numbers n1(2) = n - 1, n2(1) = m = 0 in the case of collinear orientation of the Stark-state dipolar electric moments and the interatomic axis. The cubic polynomial K3(x) in powers of reciprocal values of the principal quantum number x = 1/n and quadratic polynomial K2(y) in powers of reciprocal values of the principal quantum number squared y = 1/n2 were determined on the basis of the standard curve fitting polynomial procedure from the calculated data for C6(n). The transformation of attractive van-der-Waals force (C6 > 0) for low-energy states n < 23 into repulsive force (C6 < 0) for all higher-energy states of n ≥ 23, is observed from the results of numerical calculations based on the second-order perturbation theory for the operator of the long-range interaction between neutral atoms. This transformation is taken into account in the asymptotic formulas (in both cases of p = 2, 3) by polynomials Kp tending to unity at n → ∞ (Kp(0) = 1). The transformation from low-n attractive van-der-Waals force into high-n repulsive force demonstrates the gradual increase of the negative contribution to C6(n) from the lower-energy two-atomic states, of the A(B)-atom principal quantum numbers nA(B)′=n-Δn n′A(B) = n-Δn (where Δn = 1, 2, … is significantly smaller than n for the terms providing major contribution to the second-order series), which together with the states of n″B(A) = n+Δn nB(A)″=n+Δn make the joint contribution proportional to n12. So, the hydrogen-like manifold structure of the energy spectrum is responsible for the transformation of the power-11 asymptotic dependence C6(n) ∝ n11of the low-angular-momenta Rydberg states in many-electron atoms into the power-12 dependence C6(n) ∝ n12 for the dipolar states of the Rydberg manifold

Natural widths and blackbody radiation induced shift and broadening of Rydberg levels in magnesium ions

Theoretical analysis is presented of the natural lifetimes and blackbody-radiation (BBR)-induced shifts and widths of Rydberg states with small and large angular momenta l. Asymptotic presentations in elementary functions are derived for matrix elements of bound-bound, bound-free and threshold radiative transitions from hydrogenic-type states with large angular momenta, applicable to both hydrogen-like and many-electron atoms and ions. For states with small angular momenta two numerical methods based on the quantum defects were used and corresponding data are compared with one another and with the most reliable data of the literature. Asymptotic approximations are derived for natural lifetimes, thermal shifts and broadening of Rydberg states of small and high l and principal quantum numbers n ≫ 1

van der Waals interaction of atoms in circular Rydberg states

Analytical expressions for the constants of resolution in powers of inverse interatomic distance 1∕R are determined for the energy of the long-range interaction between two atoms A and B in their circular Rydberg states of equal principal quantum numbers nA = nB ≡ n ≫ 1, maximal angular momenta and magnetic quantum numbers lA = lB = |mA| = |mB| = n − 1 in the case of arbitrary orientation of the interatomic axis relative the axis of the m-quantization. Coefficients of the odd-power terms Ck∕Rk are derived for k = 5, 7, 9 from the first-order perturbation theory (PT) for the interaction Hamiltonian. The constant C6 of the van der Waals interaction ΔEvdW = −C6∕R6, determined from the second-order PT, is resolved into irreducible components. The combinations of irreducible components determine coefficients at Legendre polynomials and/or cosine functions describing the dependence of C6 on the angle between the interatomic and quantization axes. The asymptotic dependence on the principal quantum number C6 ∝ n12 is demonstrated analytically. Polynomials in powers of n are derived for extrapolation of calculated data to the case of arbitrary n. The transformation of attractive van der Waals force (C6 > 0) for low-energy states n < 6 into repulsive force (C6 < 0) for all higher-energy states of n ≥ 8 is demonstrated in numerical calculations