# Long-Range Atom–Ion Rydberg Molecule: A Novel Molecular Binding Mechanism

^{ST}, Institut für Quantenmaterie, Universität Ulm, 89069 Ulm, Germany

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Binding Mechanism and Properties of the Long-Range Atom–Ion Rydberg Molecules

## 3. Stability and Lifetime of the Long-Range Atom–Ion Rydberg Molecules

## 4. Production and Detection of the Long-Range Atom–Ion Rydberg Molecules

#### 4.1. Production by Photoassociation

#### 4.2. Detection by Photoionization

## 5. Prospects for Experiments with Long-Range Atom–Ion Rydberg Molecules

## 6. Conclusions and Outlook

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) Stark map level structure of a Rb atom in the vicinity of the $17P$ state as a function of the internuclear distance r between the atom and the ion. The energy reference is given by the term energy of the $17{P}_{3/2}$ state at zero electric field corresponding to the value for $r\to \infty $. Blue (gray) solid lines represent levels with $|{m}_{J}|=1/2$ ($|{m}_{J}|=3/2$), where ${m}_{J}$ is the magnetic quantum number. The purple vertical line marks a critical internuclear distance ${r}_{\mathrm{c}}$. For $r\lesssim {r}_{\mathrm{c}}$, our model breaks down (see text). (

**b**) Magnification of the region indicated by the magenta arrow in Figure 1a, showing the two outermost potential wells. The red solid horizontal lines correspond to vibrational level energies. The parameters $\Delta E$, $\Delta r$, and ${r}_{\mathrm{min}}$ are used to characterize a potential well (see text). (

**c**,

**d**) Stark map and magnification of the vicinity of the $47P$ state. Here, the energy reference is the term energy of the $47{P}_{3/2}$ state at zero electric field.

**Figure 2.**Properties of the second outermost potential wells associated with the $n{P}_{1/2}$ states of Rb and of corresponding molecular bound states as functions of the principal quantum number n. (

**a**) Depth $\Delta E$ (on a logarithmic scale). (

**b**) Width $\Delta r$. (

**c**) Position ${r}_{\mathrm{min}}$. (

**d**) Number N of vibrational bound states. (

**e**) Energy splitting $\Delta {E}_{\mathrm{v}}$ between the vibrational ground state and the first excited vibrational state (on a logarithmic scale). (

**f**) Rotational constant B for a ${}^{87}$Rb${}^{138}$Ba${}^{+}$ molecule (on a logarithmic scale).

**Figure 3.**(

**a**) Over-barrier motion. The red solid lines represent the Coulomb potential ${V}_{\mathrm{Cou}}$ as given in Equation (11) for two ionic cores located at $z=0$ and $z=80\phantom{\rule{0.222222em}{0ex}}\mathrm{nm}$ on the z-axis (i.e., $r=80\phantom{\rule{0.222222em}{0ex}}\mathrm{nm}$) and an electron at position z. The black solid horizontal lines correspond to the energy of the unperturbed Rydberg level for $n=17$ and $n=22$, respectively. (

**b**) Comparison of the critical internuclear distance ${r}_{\mathrm{c}}$ to the internuclear distance ${r}_{\mathrm{min}}$ where the minimum of the second outermost potential well associated with $n{P}_{1/2}$ states is located, as a function of n.

**Figure 4.**Comparison of wave functions. (

**a**) Upper panel: Potential energy curves in the region of the two outermost potential wells associated with the $17{P}_{1/2}$ (on the left; see also Figure 1b) and the $27{P}_{1/2}$ (on the right) states. Blue (gray) solid lines indicate levels with $|{m}_{J}|=1/2$ ($|{m}_{J}|=3/2$). Black dashed vertical lines mark internuclear distances below which the potential energy landscape is not plotted. The energy reference is the term energy of the $17{P}_{3/2}$ and $27{P}_{3/2}$ state, respectively, at zero electric field ($r\to \infty $). Middle panel: The blue solid lines show the potential energy curve for a Rb $5{S}_{1/2}$ atom colliding with a Ba${}^{+}$$6{S}_{1/2}$ ion within the partial wave ${l}^{\prime}=0$. We choose the ${\left(1\right)}^{3}{\mathsf{\Sigma}}^{+}$ state to represent the short-range part for $r\lesssim 4\phantom{\rule{0.222222em}{0ex}}\mathrm{nm}$ (see text). Here, the energy reference corresponds to the atomic asymptote of the two collision partners. The black, magenta, and ocher solid lines are scattering wave functions for atom–ion collision energies E of $(1,\phantom{\rule{0.166667em}{0ex}}0.1,\phantom{\rule{0.166667em}{0ex}}0.01)\phantom{\rule{0.222222em}{0ex}}\mathrm{mK}\times {k}_{\mathrm{B}}$, respectively. Their amplitudes are scaled for better visibility and are given in arbitrary units. Lower panel: The same as the middle panel but for ${l}^{\prime}=20$. For the blue solid lines, the angular momentum potential is included. (

**b**,

**c**) Magnifications of parts of Figure 4a. For simplicity, in the upper panels, only the second outermost potential wells associated with the states $17{P}_{1/2}$ and $27{P}_{1/2}$ are considered. The red solid lines are the wave functions of vibrational levels with ${l}^{\prime}=0$ (presented by using a scaling factor and given in arbitrary units).

**Figure 5.**Examples for coupling of molecular levels. Shown are the potential energy curves in the region around the two outermost molecular potential wells associated with the $27{P}_{1/2}$ state of Rb. Blue (gray) solid lines indicate levels with $|{m}_{J}|=1/2$ ($|{m}_{J}|=3/2$). Here, the reference for zero energy is the term energy of the $27{P}_{3/2}$ state at zero electric field ($r\to \infty $). The red solid lines correspond to wave functions of vibrational levels with ${l}^{\prime}=0$. Their amplitudes are scaled for better visibility and are given in arbitrary units. The green arrow illustrates a transition between two neighboring vibrational states within the same potential well, which can be driven by a radio-frequency field. The purple vertical arrow represents a microwave transition between two vibrational states in different potential wells. The cyan vertical arrow indicates a transition from a bound state towards a repulsive potential energy curve.

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**MDPI and ACS Style**

Deiß, M.; Haze, S.; Hecker Denschlag, J.
Long-Range Atom–Ion Rydberg Molecule: A Novel Molecular Binding Mechanism. *Atoms* **2021**, *9*, 34.
https://doi.org/10.3390/atoms9020034

**AMA Style**

Deiß M, Haze S, Hecker Denschlag J.
Long-Range Atom–Ion Rydberg Molecule: A Novel Molecular Binding Mechanism. *Atoms*. 2021; 9(2):34.
https://doi.org/10.3390/atoms9020034

**Chicago/Turabian Style**

Deiß, Markus, Shinsuke Haze, and Johannes Hecker Denschlag.
2021. "Long-Range Atom–Ion Rydberg Molecule: A Novel Molecular Binding Mechanism" *Atoms* 9, no. 2: 34.
https://doi.org/10.3390/atoms9020034