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arXiv:1501.06014 · New Journal of Physics

A comparative analysis of binding in ultralong-range Rydberg molecules

Christian Fey, Markus Kurz, Peter Schmelcher, Seth T. Rittenhouse, H. R. Sadeghpour

Published May 7, 2015• 35 citations
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Abstract

We perform a comparative analysis of different computational approaches employed to explore the electronic structure of ultralong-range Rydberg molecules. Employing the Fermi pseudopotential approach, where the interaction is approximated by an s -wave bare delta function potential, one encounters a non-convergent behavior in basis set diagonalization. Nevertheless, the energy shifts within the first order perturbation theory coincide with those obtained by an alternative approach relying on Green's function calculation with the quantum defect theory. A pseudopotential that yields exactly the results obtained with the quantum defect theory, i.e. beyond first order perturbation theory, is the regularized delta function potential. The origin of the discrepancies between the different approaches is analytically explained.

DOI
10.1088/1367-2630/17/5/055010

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We observe weakly-bound $Π$-symmetry electronic states in the spectroscopy of $^{87}$Rb$(nP_{3/2})$+$^{87}$Rb($5S_{1/2}$) ultralong-range Rydberg molecules. We detect these molecules in Rydberg states having principal quantum number $13\le n \le 16$. Their $Π$-state character is unambiguously identified via their observed multiplet structure: the $2F+1$ magnetic sublevels of the ground-state rubidium atom separate, as in the Zeeman effect, because of the spin-spin coupling between the Rydberg and valence electrons. We find a rapid decrease in the molecular binding energy $\propto (n-μ_{P_{3/2}})^{-11}$, where $μ_{P_{3/2}}$ is the quantum defect, indicating that the low-$n$ regime of Rydberg states is ideally suited for studies of $Π$-symmetry molecules. Our observations are in good agreement with Green's function-based calculations for $14\le n\le 16$, with poorer agreement for $n=13$ hinting at the beginning of a breakdown of the Fermi pseudopotential approach at low $n$.

arXiv:2608.07447•Aug 7, 2026•0 citations
We perform a comparative analysis of different computational approaches employed to explore the electronic structure of ultralong-range Rydberg molecules. Employing the Fermi pseudopotential approach, where the interaction is approximated by an s -wave bare delta function potential, one encounters a non-convergent behavior in basis set diagonalization. Nevertheless, the energy shifts within the first order perturbation theory coincide with those obtained by an alternative approach relying on Green's function calculation with the quantum defect theory. A pseudopotential that yields exactly the results obtained with the quantum defect theory, i.e. beyond first order perturbation theory, is the regularized delta function potential. The origin of the discrepancies between the different approaches is analytically explained.
We investigate polyatomic ultralong-range Rydberg molecules consisting of three ground-state atoms bound to a Rydberg atom via sss- and ppp-wave interactions. By employing the finite basis set representation of the unperturbed Rydberg electron Green's function we reduce the computational effort to solve the electronic problem substantially. This method is subsequently applied to determine the potential energy surfaces of triatomic systems in electronic sss- and ppp-Rydberg states. Their molecular geometry and resulting vibrational structure are analyzed within an adiabatic approach that separates the vibrational bending and stretching dynamics. This procedure yields information on the radial and angular arrangement of the nuclei and indicates in particular that kinetic couplings between bending and stretching modes induce a linear structure in triatomic l=0l=0l=0 ultralong-range Rydberg molecules.
In this work the existence of a novel type of molecule is shown whose binding is based solely on electron-atom scattering. These molecules are composed of a Rydberg atom and a ground state atom - here Rb(ns) and Rb(5s) - where the binding arises from scattering of the Rydberg electron from the ground state atom. The ground state atom is bound to the Rydberg wavefunction, which has radii on the order of 100nm, and thus these molecules are of long-range. Although the theoretical description of this interaction goes back to the famous work of Fermi in the year 1934 which was expanded in the following decades by Omont and Greene, the experimental proof of their existence was still missing. Up to now, experimental evidence for the ultralong-range Rydberg molecules was found only in spectral line broadenings. The present experiments have demonstrated the direct photoassociation of these ultralong-range Rydberg molecules for Rb(ns)-Rb(5s) triplet states from a cold and dense sample of rubidium 87 atoms for a range of principal quantum numbers, n, between 34 and 40. The high-resolution photoassociation spectra allowed for the observation of several vibrational states separated by only a few Megahertz. Based on the model by Greene the vibrational ground states v=0 are assigned and the dependence of their binding energy on the principal quantum number, n, is used to extract the triplet s-wave scattering length for electron-Rb(5s) collisions from the data. The remarkably good agreement between experiment and theory for the vibrational ground state v=0 over a range of principal quantum numbers attests to the accuracy of Fermi's original pseudopotential approach in describing interactions in excited multi-atom systems. Aside from this good agreement for the state v=0, the measurements also show that contributions from p-wave scattering cannot be neglected, as a number of excited vibrational states is observed in the experiment that are not predicted from the theory assuming only s-wave scattering. Calculations of Li, Pohl and coworkers have shown that a full solution of the electron-atom scattering is necessary for the assignment of all vibrational states. This work finds the molecular bound states to be very sensitive to both s-wave as well as p-wave electron-atom scattering and the Rydberg molecules turn out to be an accurate experimental platform to study electron-atom collisions in a previously inaccessible ultralow-energy regime. In general, the ultralong-range Rydberg molecules investigated here for rubidium are expected for all species with negative scattering length for electron-atom interaction, e. g. the other alkali atoms. Rydberg atom and ground state atom do not even have to be of the same chemical element, the only requirement is a negative scattering length of the ground state atom for electron scattering. Moreover, the binding force of the Rydberg electron is not restricted to a single ground state atom and molecules with more than one ground state atom can be created. The fact that these exotic molecules have huge bond lengths but nevertheless a vibrational spectrum with several states make a new type of ultracold Rydberg chemistry feasible. In the future, larger polymers or even heteronuclear molecules can be realized, where the number of atoms, the constituent atomic elements and the addressed Rydberg state allow to precisely select the properties of the long-range molecule.
Divalent atoms have emerged as powerful alternatives to alkalis in ultracold atom platforms, offering unique advantages arising from their two-electron structure. Among these species, ytterbium (Yb) is especially promising, yet its anionic properties and its Rydberg spectrum remain comparatively unexplored. In this work, we perform a first and comprehensive experimental and theoretical investigation of ultralong-range Rydberg molecules (ULRMs) of 174^{174}174Yb in 6sns 1S06sns\,^1S_06sns1S0​ Rydberg states across nearly two decades in principal quantum number nnn and three orders of magnitude in molecular binding energy. Using the Coulomb Green's function formalism, we compute Born-Oppenheimer molecular potentials describing the Rydberg atom in the presence of a ground-state perturber and achieve quantitative agreement with high-resolution molecular spectra. This enables the extraction of low-energy electron-Yb scattering phase shifts, including the zero-energy sss-wave scattering length and the positions of two spin-orbit split ppp-wave shape resonances. Our results provide strong evidence that the Yb−^{-}− anion exists only as a metastable resonance.% We additionally show the sensitivity of ULRM spectra to the atomic quantum defects, using this to determine the quantum defect of the 6s23f 1F36s23f\, ^1F_36s23f1F3​ state. Together, these findings establish Yb ULRMs as a powerful probe of electron-Yb interactions and lay essential groundwork for future Rydberg experiments with divalent atoms.
An operator that generates an approximate symmetry of long-range Rydberg molecules (LRRMs) formed by two alkali atoms, one in a Rydberg state and the other in the ground state, is identified. This is first done by evaluating the natural orbitals associated with a variational calculation of the binding wave function within the Born-Oppenheimer description of the molecule including s and p Fermi pseudopotential and the hyperfine structure energy terms. The resulting orbitals with the highest occupation number are shown to be identical to those obtained by a perturbative model for high angular momentum—trilobite and butterfly—LRRMs. Whenever the slight dependence of the quantum defects of the Rydberg electron on its total momentum j⃗=ℓ⃗+s⃗1 can be neglected, the symmetry operator of the high angular momentum LRRMs orbitals is identified as the sum of the spin of the Rydberg electron s⃗1 , spin of the valence electron s⃗2 , and the spin of nucleus i⃗ of the ground-state atom, N⃗=s1⃗+s2⃗+i⃗ . The spin orbitals that diagonalize N⃗ define compact basis sets for the description of LRRMs beyond the aforementioned approximations. The matrix elements of the Hamiltonian in these basis sets have simple expressions, so that the relevance of triplet and singlet contributions can be directly estimated. The expected consequences of this approximate spin-symmetry on the spectra of LRRMs are briefly described. Published by the American Physical Society 2024
We observe weakly-bound ΠΠΠ-symmetry electronic states in the spectroscopy of 87^{87}87Rb(nP3/2)(nP_{3/2})(nP3/2​)+87^{87}87Rb(5S1/25S_{1/2}5S1/2​) ultralong-range Rydberg molecules. We detect these molecules in Rydberg states having principal quantum number 13≤n≤1613\le n \le 1613≤n≤16. Their ΠΠΠ-state character is unambiguously identified via their observed multiplet structure: the 2F+12F+12F+1 magnetic sublevels of the ground-state rubidium atom separate, as in the Zeeman effect, because of the spin-spin coupling between the Rydberg and valence electrons. We find a rapid decrease in the molecular binding energy ∝(n−μP3/2)−11\propto (n-μ_{P_{3/2}})^{-11}∝(n−μP3/2​​)−11, where μP3/2μ_{P_{3/2}}μP3/2​​ is the quantum defect, indicating that the low-nnn regime of Rydberg states is ideally suited for studies of ΠΠΠ-symmetry molecules. Our observations are in good agreement with Green's function-based calculations for 14≤n≤1614\le n\le 1614≤n≤16, with poorer agreement for n=13n=13n=13 hinting at the beginning of a breakdown of the Fermi pseudopotential approach at low nnn.