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Quantum Electronics

Interatomic interactions and thermally induced shifts and broadenings of energy levels of atoms in circular Rydberg states

Aleksandr A. Kamenski, Vitalii D Ovsiannikov, Igor L. Glukhov

Published May 8, 2019• 9 citations
Publisher

Abstract

Abstract Analytical expressions are obtained for the constants determining the energy of the long-range (at large distances exceeding the LeRoy radius R LR ≈ 4n 2) polarisation interaction of the multipole electric moments of two atoms in identical Rydberg states with large principal (n) and maximum orbital (l) and magnetic (m) quantum numbers (|m| = l = n − 1). Asymptotic expansions in powers of n are found for the components of the van der Waals interaction tensor, which determine the dependence of the dispersion energy on the orientation of the interatomic axis. The spontaneous and thermally induced broadenings of the energy levels of circular states are represented analytically as functions of the principal quantum number and temperature. It is found that analytical formulae for the sum of spontaneous and thermally induced line widths in the region of high temperatures and large principal quantum numbers (assuming n 3 k B T >> 1) do not contain a temperature-independent contribution of the spontaneous part of the broadening. Closed analytical expressions are also obtained for the Farley − Wing function and for the n- and T-dependent corrections to the asymptotic behaviour of the thermally induced shift.

DOI
10.1070/qel17000

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Quantum Electronics•May 19, 2021•1 citation
Abstract Analytical expressions are obtained for the constants determining the energy of the long-range (at large distances exceeding the LeRoy radius R LR ≈ 4n 2) polarisation interaction of the multipole electric moments of two atoms in identical Rydberg states with large principal (n) and maximum orbital (l) and magnetic (m) quantum numbers (|m| = l = n − 1). Asymptotic expansions in powers of n are found for the components of the van der Waals interaction tensor, which determine the dependence of the dispersion energy on the orientation of the interatomic axis. The spontaneous and thermally induced broadenings of the energy levels of circular states are represented analytically as functions of the principal quantum number and temperature. It is found that analytical formulae for the sum of spontaneous and thermally induced line widths in the region of high temperatures and large principal quantum numbers (assuming n 3 k B T >> 1) do not contain a temperature-independent contribution of the spontaneous part of the broadening. Closed analytical expressions are also obtained for the Farley − Wing function and for the n- and T-dependent corrections to the asymptotic behaviour of the thermally induced shift.
Abstract Based on the rules for summing oscillator-strength moments, we derive analytical expressions for environmentally (thermally) induced energy-level shifts and broadenings of circular Rydberg states with maximum orbital (l) and magnetic (m) quantum numbers, l = |m| = n – 1. The formulae for the energy of interaction with blackbody radiation are presented in the form of series expansion in even powers of the small parameter η = Z 2/(n 3 k B T) ≪ 1 in the region of high temperatures T and principal quantum numbers n (Z is the charge of the residual ion). The expression for the thermally induced width contains a negative temperature-independent term coinciding in absolute value with the spontaneous width, so that the temperature-independent term is absent in the expression for the sum of the spontaneous and thermally induced widths. A similar expansion in powers of η for the thermally induced shift also contains a temperature-independent term proportional to 1/n 6.
With the use of second-order perturbation theory in the long-range interatomic interaction for the degenerate states of two Rydberg atoms we have obtained a general formula for the dependence of atomic interaction energy on the interatomic distance R in the presence of the Förster resonance. Inside of the ‘Förster sphere’ ( R < R F ) this dependence transforms to the formula for electric dipole interaction energy Δ E d − d = C 3 / R 3 and for R > R F it transforms to the formula for the van der Waals interaction energy Δ E VdW = − C 6 / R 6 . The van der Waals constant C 6 is represented as an expansion in terms of irreducible components which define the dependence on the interatomic axis orientation relative to the quantisation axis of projections M of the total angular momentum J . The numerical values of the irreducible components of tensor C 6 were calculated for rubidium atoms in the same Rydberg states | nlJM 〉 with large quantum numbers n . We present the calculated resonance interaction energy of two rubidium atoms in the states |43D 5/2 M 〉, whose total energy exceeds by only 8 MHz the total energy of one of the atoms in the state |45P 3/2 M 〉 and of the other in the state |41F 7/2 M 〉.
Abstract Analytical expressions for spontaneous and induced by blackbody radiation energy shifts and widths of atomic circular states are derived. The features of asymptotic behavior of the total width and energy shifts for highly excited Rydberg circular states in the blackbody radiation field are described. Approximate equations are proposed for simple evaluations of energy characteristics and an absence of temperature-independent spontaneous contribution to the total energy width in a range of high temperatures and principal quantum numbers is demonstrated.
Irreducible components determining the dependence of the van der Waals coefficient C 6 ( n l J M ) on the angle θ between the interatomic and the quantisation axes of two Rb atoms in their identical Rydberg states | n l J M ⟩ are evaluated with account of the most contributing terms of the spectral resolution for the bi-atomic Green’s function. Asymptotic polynomials in powers of the Rydberg-state principal quantum number n are derived for the C 6 irreducible components. Numerical values of the polynomial coefficients are determined for Rb atoms in their n 2 S 1 / 2 , n 2 P 1 / 2 , 3 / 2 , n 2 D 3 / 2 , 5 / 2 and n 2 F 5 / 2 , 7 / 2 Rydberg states of arbitrary high n . The transformation of the van-der-Waals interaction law − C 6 / R 6 into the dipole–dipole law C 3 / R 3 in the case of close two-atomic states (the Förster resonance) is considered. Numerical values, the dependences on the magnetic quantum numbers M and on the angle θ of the constant C 3 are determined together with the ranges of interatomic distances R , where the interaction law R − 6 transforms into the law of R − 3 .
Analytical expressions for spontaneous and induced by blackbody radiation energy shifts and widths of atomic circular states are derived. The features of asymptotic behavior of the total width and energy shifts for highly excited Rydberg circular states in blackbody radiation field are described. Approximate equations are proposed for simple evaluations of energy characteristics and an absence of temperature-independent spontaneous contribution to the total energy width in a range of high temperatures and principal quantum numbers is demonstrated.
Abstract We present the derivation of simple analytical expressions for calculating the widths of Rydberg states with small orbital angular momenta in Zn+, Cd+, Hg+ ions of group IIb of the periodic system of elements induced by blackbody radiation (BBR) at ambient temperatures from 100 to 3000 K. The probabilities of radiative transitions from excited nS, nP, nD and nF states to all dipole-accessible states of ions are calculated. The wave functions of the quantum defect (QD) method for the initial and final states of the Rydberg electron are used to calculate the amplitudes of transitions between bound states. The dependences of the probabilities of induced decays and excitations on the BBR temperature, the principal and orbital quantum numbers of the Rydberg ion are determined. Analytical expressions are derived for numerical estimates of the contributions of the probabilities of thermally induced decays and excitations to the width of the Rydberg energy level. The numerical values of the coefficients of interpolation polynomials representing asymptotic expansions in powers of the principal quantum number n for the relative probabilities of decays and excitations of Rydberg states with large values of n and small orbital angular momenta l = 0, 1, 2, 3 are calculated.