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Physical Review A

Adiabatic energy levels and electric dipole moments of Rydberg states of Rb2 and Cs2 dimers

A.A. Khuskivadze, M. I. Chibisov, I. I. Fabrikant

Published Oct 15, 2002• 117 citations
Publisher

Abstract

We calculate potential energy curves for heavy alkali-metal dimers, ${\mathrm{Rb}}_{2}$ and ${\mathrm{Cs}}_{2}$ in which one of the atoms is in a highly excited Rydberg states. The method combines numerical integration of coupled equations, describing interaction of electron with the ground-state atom in the field of the Coulomb core of the Rydberg atom, with subsequent matching of the obtained wave function with the Coulomb Green's function in the form of the Kirchhoff integral. The spin-orbit interaction for the Rydberg electron is also included. The results show the existence of several groups of states. Most interesting of them are dominated either by the ${}^{3}S$ symmetry near the ground-state atom or by the ${}^{3}{P}_{J}$ symmetry, $J=0,1,2.$ All states, except the ${}^{3}{P}_{1}$ state, exhibit oscillatory dependence of energy on the internuclear distance that can support long-range molecular Rydberg states [e.g., ``trilobite states'' for the ${}^{3}S$ symmetry, Greene et al., Phys. Rev. Lett. 85, 2458 (2000)]. These states possess large diagonal and transition dipole moments which are expressed analytically in terms of the Coulomb wave functions and calculated in a broad range of internuclear separations.

DOI
10.1103/physreva.66.042709

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We calculate potential energy curves for heavy alkali-metal dimers, Rb2{\mathrm{Rb}}_{2}Rb2​ and Cs2{\mathrm{Cs}}_{2}Cs2​ in which one of the atoms is in a highly excited Rydberg states. The method combines numerical integration of coupled equations, describing interaction of electron with the ground-state atom in the field of the Coulomb core of the Rydberg atom, with subsequent matching of the obtained wave function with the Coulomb Green's function in the form of the Kirchhoff integral. The spin-orbit interaction for the Rydberg electron is also included. The results show the existence of several groups of states. Most interesting of them are dominated either by the 3S{}^{3}S3S symmetry near the ground-state atom or by the 3PJ{}^{3}{P}_{J}3PJ​ symmetry, J=0,1,2.J=0,1,2.J=0,1,2. All states, except the 3P1{}^{3}{P}_{1}3P1​ state, exhibit oscillatory dependence of energy on the internuclear distance that can support long-range molecular Rydberg states [e.g., ``trilobite states'' for the 3S{}^{3}S3S symmetry, Greene et al., Phys. Rev. Lett. 85, 2458 (2000)]. These states possess large diagonal and transition dipole moments which are expressed analytically in terms of the Coulomb wave functions and calculated in a broad range of internuclear separations.
Adiabatic wavefunctions and dipole moments of Rydberg states perturbed by a neutral atom have been investigated. Wavefunctions are equal to sums of products of Coulomb wavefunctions and differ very substantially from any single Coulomb wavefunction. Using the Kirchhoff-integral method (Fabrikant I I 1993 J. Phys. B: At. Mol. Opt. Phys. 26 2533) we calculate the potential curves of two adiabatic Rydberg states for the Rb*( n ≃30) + Rb(5s) system: one dominated by the 3 S symmetry near the ground-state Rb atom, and the other dominated by the 3 P symmetry.
The adiabatic \ensuremath{\Sigma} electronic terms of highly excited states of the He--alkali-metal atom systems are determined as the roots of a transcendental equation whose only required inputs are the alkali-metal quantum defects, the ${e}^{\mathrm{\ensuremath{-}}}$-He s-wave phase shifts and, of course, the internuclear distance R. The results obtained for relatively low excited states are in reasonable agreement with more elaborate calculations. It is found that the notion of adiabatic quantum defects requires R(a.u.)\ensuremath{\lesssim}2n, n being the principal quantum number. The method is generalized to the case where a small external electric field scrE is present. Then the energy levels strongly depend on the angle between scrE and the internuclear axis. Numerical applications to He-H are presented.
We introduce a computational method developed for study of long-range molecular Rydberg states of such systems that can be approximated by two electrons in a model potential of the atomic cores. Only diatomic molecules are considered. The method is based on a two-electron RRR-matrix approach inside a sphere centered on one of the atoms. The wave function is then connected to a Coulomb region outside the sphere via a multichannel version of the Coulomb Green's function. This approach is put into a test by its application to a study of Rydberg states of the hydrogen molecule for internuclear distances RRR from 20 to 400 bohrs and energies corresponding to nnn from 3 to 22. The results are compared with previous quantum chemical calculations (lower quantum numbers nnn) and computations based on contact-potential models (higher quantum numbers nnn).
The adiabatic potential energy curves of Rb2{\mathrm{Rb}}_{2}Rb2​ in the long-range Rydberg electronic states are calculated using the two-electron RRR-matrix method [M. Tarana and R. \ifmmode \check{C}\else \v{C}\fi{}ur\'{\i}k, Phys. Rev. A 95, 042515 (2017)] for the intermediate internuclear separations between 37 and 200 a.u. The results are compared with the zero-range models to find a region of the internuclear distances where Fermi's pseudopotential approach provides accurate energies. A finite-range potential model of the atomic perturber is used to calculate the wave functions of the Rydberg electron and their features specific for the studied range of internuclear distances are identified.
The work presented in this dissertation represents a comprehensive study on long-range interactions between alkali Rydberg atoms. As a consequence of their size, Rydberg atoms possess very large dipole moments, making them highly sensitive to external fields and to other Rydberg atoms. The strong dipole-dipole interaction between two Rydberg atoms results in a mixing of their angular momentum characters, which is the cornerstone of the work presented here. We describe the ℓ-mixing between various local molecular asymptotes through a residual Coulomb potential, assuming no electron cloud overlap. We then directly diagonalize an interaction Hamiltonian consisting of these Rydberg-Rydberg interactions and spin-orbit (fine structure) coupling in the Hund's case (c). In this manner, we calculate potential energy curves for various molecular symmetries and find that some of these curves exhibit deep potential wells, capable of supporting very extended bound molecular states ( macrodimers). We analyze the specific structure of the ℓ-mixing for these potential wells and derive convenient n-scaling relations for both their equilibrium separations Re and well depths De. We also explore various properties of the macrodimers, including their response to small electric fields and stability with respect to predissociation. We note throughout the dissertation that the detection of such extended dimers could help progress studies in a variety of areas such as exotic, ultracold chemistry and quantum information and computing. To facilitate detection, we propose a scheme to form and study the macrodimers via photoassociation (PA). We present calculated PA lineshapes from two different electronic ground states based on a harmonic trapping model. We find that both PA signals vary significantly with the laser detunings, indicating that different ℓ-characters could be probed at different equilibrium lengths. Finally, we extend the two-body formalism to the case of three interacting Rydberg atoms and present preliminary results for triply excited 58p rubidium atoms along a common axis. We provide a few examples of surface plots corresponding to different local asymptotes, each with a unique three-dimensional "landscape" and discuss the relevance of the physical features. The highlight of our analysis is the appearance of a potential well, which indicates the formation of a linear trimer. ^
We have studied the long-range interaction potentials of the heteronuclear alkali-metal-atom dimers in the case when one of the atoms is in the first excited state (nA{n}_{A}nA​P) and the other is in the ground state (nB{n}_{B}nB​S). The presentation of the general expressions for the interaction energies in the Hund's case (c) is followed by the calculation of the corresponding C6{C}_{6}C6​ constants for all heteronuclear alkali-metal-atom pairs. In the case of heavier perturbers (Rb and Cs) the C6{C}_{6}C6​ constants are found to be significantly influenced by the perturber's fine structure. The van der Waals interaction in the K-Rb system is exceptionally strong resulting in C6{C}_{6}C6​ constants considerably larger than in all other alkali-metal-atom heteronuclear systems. The intermediate-coupling region (ICR) is defined as the region of internuclear separation R where the interaction energy is of the order of the atomic spin-orbit splitting, while the exchange energy is still negligible. Analysis of the general form of the energy matrices in the ICR shows that the homonuclear alkali-metal-atom dimers can be treated as a special case. In the large-R limit, the ICR results reduce to those obtained in Hund's case (c). As an example, we have calculated the ICR interaction potentials of the K-Rb system, taking into account only the leading dipole-dipole interaction. Finally we have discussed the construction of the absorption coefficient in the wings of the resonance lines, both in the impact and in the quasistatic approximation.