## Bonding's outer limit

In a Rydberg state, an atom has been very nearly, but not quite, ionized. This puts the electron relatively far from the nucleus, and two atoms in such a state can thus form a rather long-range bond. Hollerith *et al.* observed this phenomenon in fine detail by exciting pairs of ultracold rubidium atoms along the diagonal of an optical lattice. The authors resolved the vibrational state structure spectroscopically and showed that the Rydberg dimers manifested bond lengths exceeding 700 nanometers.

*Science*, this issue p. 664

## Abstract

The subnanoscale size of typical diatomic molecules hinders direct optical access to their constituents. Rydberg macrodimers—bound states of two highly excited Rydberg atoms—feature interatomic distances easily exceeding optical wavelengths. We report the direct microscopic observation and detailed characterization of such molecules in a gas of ultracold rubidium atoms in an optical lattice. The bond length of about 0.7 micrometers, comparable to the size of small bacteria, matches the diagonal distance of the lattice. By exciting pairs in the initial two-dimensional atom array, we resolved more than 50 vibrational resonances. Using our spatially resolved detection, we observed the macrodimers by correlated atom loss and demonstrated control of the molecular alignment by the choice of the vibrational state. Our results allow for rigorous testing of Rydberg interaction potentials and highlight the potential of quantum gas microscopy for molecular physics.

A quantitative determination of the structure of molecules is an essential goal of physical chemistry and is crucial to revealing and understanding their properties. The high level of quantum control and the ultracold temperatures achieved in atomic physics provide tools to study molecules and their structure (*1*). Prominent examples include the observations of weakly bound Feshbach molecules (*2*, *3*), the controlled photoassociation of individual molecules in a microtrap (*4*), molecules comprising ground-state atoms bound to a highly excited Rydberg atom (*5*–*7*), or pure long-range molecules (*8*, *9*) that are bound purely electrostatically. The binding mechanism of the latter, where the electron orbitals of the constituents do not overlap, has also been predicted to pertain between two Rydberg atoms. These “Rydberg macrodimers” (*10*–*14*) are truly remarkable in their macroscopic bond lengths, which are 10,000 times those of usual diatomic molecules and thus reach typical interparticle distances in magneto-optical traps (*15*), optical lattices (*16*), or optical tweezers (*17*, *18*). Their enormous size not only enables direct optical access to individual constituents, but also allows for the controlled binding of two atoms optically pinned at the correct distance. First signatures of Rydberg macrodimers have been observed in systems of laser-cooled atoms by spectroscopy (*15*) and pulsed-field ionization (*19*), but a vibrationally and spatially resolved detection has been lacking so far.

Here, we present a precise study of Rydberg macrodimers starting from ground-state ^{87}Rb atoms deterministically arranged in an optical lattice. We probed the vibrational levels by two-photon spectroscopy, resolving more than 50 excited vibrational resonances. The observed spectrum agrees quantitatively with ab initio calculations of Rydberg interaction potentials (*20*–*22*), providing a stringent test for their accuracy. Using the site-resolved detection and single-atom sensitivity of our quantum gas microscope (*23*, *24*), we identified the macrodimer signal microscopically as a loss of pairs of ground-state atoms at a distance of a bond length. Furthermore, we controlled the spatial orientation of the photoassociated molecules by the parity of the vibrational wave function and the polarization of the excitation laser.

Rydberg interaction potentials are of van der Waals type for asymptotically large interatomic separations *R *(*25*), but the situation is more complicated at smaller distances. In a generic situation, repulsive interactions may increase the energy of a lower-lying pair state, _{ν}(*R*) (*10*, *11*, *15*), where ν denotes the vibrational quantum number of the nuclear motion. In our experiment, we chose such an avoided crossing of two *1*, *20*). For large distances, the selected pair states transform into the states *26*), which can be optically coupled from the ground state |*g*, *g*〉 by a two-photon transition. Here, |*g*〉 = |5*S*_{1/2}*F* = 2, *m*_{F} = 0〉 and the bond length of this macrodimer state is predicted to be 720 nm, close to the diagonal lattice spacing of *R*_{in} = 752 nm in our optical lattice. With the atoms initially prepared in the motional ground state of the optical lattice, this coincidence of length scales results in a strong optical coupling due to the large wave function overlap.

Our experiments started with a two-dimensional atomic Mott insulator of ^{87}Rb with a lattice filling of 94 ± 1% in the atomic ground state. The atoms were pinned in a deep optical lattice with a root mean square (RMS) width σ_{lat} = 68 nm of the motional ground state in the atomic plane (Fig. 1B) with a temperature below the on-site trapping frequency. The molecules were photoassociated by an ultraviolet (UV) Rydberg excitation laser at a wavelength of 298 nm propagating along the diagonal direction of the optical lattice, with linear polarization aligned either in or orthogonal to the lattice plane. Our typical optical Rabi coupling from the state |*g*〉 to |*e*〉 was Ω/2π = 1.2 ± 0.1 MHz. We detected the excited macrodimers as missing pairs of ground-state atoms as these were ejected from the optical lattice very efficiently as a result of kinetic energy release in the macrodimer decay. The remaining ground-state atoms were then imaged with near-unity fidelity using a quantum gas microscope (Fig. 1D).

We first aimed to identify the presence of the bound macrodimers by their spectroscopic fingerprint. To this end, we illuminated the atomic ensemble for *t*_{UV} = 100 ms with the UV light polarized in the atomic plane for varying detuning Δ from the bare Rydberg state |*e*〉. In each step, we swept the optical frequency of the excitation laser by 480 kHz during the illumination time to ensure coverage of the full spectral region between neighboring data points. We observed the coupling to the bare Rydberg state |*e*〉 as a very broad saturated loss resonance (Fig. 2). At negative detunings, the resonance features an asymmetric broadening due to coupling to attractively interacting pair-state potentials (*27*). At positive detunings around Δ/2π = 180 MHz, the first macrodimer bound state becomes two-photon resonant. A nonsaturated high-resolution spectrum of this line (Fig. 2C) features a full width at half maximum of 139 ± 5 kHz, which is of the same order as the measured linewidth of the bare Rydberg resonance (*28*). For higher vibrational resonances, we observed a reduction of the line strength, which we attribute to a combination of increased intermediate-state detuning and reduced overlap of the spatial wave functions. In addition, we observed a suppression of the excitation to odd vibrational states ν due to the approximate odd parity symmetry of the molecular wave functions with respect to the equilibrium distance. This is consistent with the Franck-Condon principle, which predicts the coupling to be proportional to the overlap integrals of the broad initial and the tightly confined final spatial wave functions Φ_{g}(*R*) and Φ_{ν}(*R*) (Fig. 1B). However, a closer inspection of the experimental data shows that this simple picture needs to be refined. Repeating the spectroscopy with orthogonal polarization (so that the optical electric field oscillates out of the atomic plane) results in a suppression of the line strength of the even lines.

Our microscopic access provides direct in situ information about the spatial alignment of the associated molecules and valuable insights into the underlying coupling mechanism (*26*, *29*, *30*). We compared different molecular lines by illuminating the cloud with UV light, resonant with a given vibrational state ν, until the filling of the lattice decreased to roughly 87 ± 1%. For a quantitative analysis, we evaluated spatially averaged density-density correlations

The key to understanding this striking alternation in the orientation of the molecules is the interplay of electronic and motional degrees of freedom. The Born-Oppenheimer wave function of the macrodimer can be expressed as *11*, *12*) is dominated by the state *26*, *29*). Because of the change of the pair state amplitudes around the potential minimum (Fig. 1B), *g*^{(2)}(1, 1). For the perpendicular case, the sign of *g*^{(2)}(1, –1). Moreover, for the measurement with light polarized out of plane (Fig. 1A), the spatial signal exhibits isotropic correlations because in that case the polarization is perpendicular to both lattice diagonals.

Although most of the line positions of the measured spectrum shown in Fig. 2 agree with the theoretical model, we find deviations for low-lying oscillator states. A finer scan of that region is shown in Fig. 4A. These deviations originate from a third pair state *D*_{3/2}, 37*P*_{1/2}〉 intersecting the binding potential around the potential minimum. At the degeneracy point, the weak dipole-quadrupole coupling between the intersecting potentials opens another gap energetically comparable to the vibrational energy. As a consequence, a separation of the vibrational motion and interatomic interaction is no longer possible. For the theory used to describe the coarse vibrational structure shown in Fig. 2, we only accounted for the crossing formed by the coupling between *n* = 36, where such an additional crossing is absent. In this case, we observed a pure harmonic oscillator–like spectrum in excellent agreement with the calculations. The remarkably high sensitivity of the measured line structure to even weak modifications of the interaction potentials underlines the promise of Rydberg macrodimer spectroscopy for benchmarking Rydberg interaction potentials. This also holds for Rydberg interactions in the presence of applied magnetic fields, where accurate calculations are more difficult (*26*).

In the future, the coupling to macrodimers could be used to realize quantum gates at well-defined qubit distance or to enhance Rydberg dressing schemes (*28*, *31*), where the pair state admixture of the strongly interacting doubly excited state is enhanced relative to the singly excited intermediate states. The strongly spatially dependent loss revealed in the correlation measurements and also by modulating the initial atom distribution (*26*) could be used to engineer dissipatively stabilized few- or many-body states (*32*, *33*). Furthermore, the approach demonstrated here can readily be extended to study multi-atom bound states (*34*, *35*). Finally, bringing the coupling rate to the macrodimers closer to the decay rate of the individual Rydberg atoms may allow for the observation of novel many-body physics arising from spatial constraints and coherent interactions.

## Supplementary Materials

science.sciencemag.org/content/364/6441/664/suppl/DC1

Materials and Methods

Figs. S1 to S9

Tables S1 and S2

This is an article distributed under the terms of the Science Journals Default License.

## References and Notes

**Acknowledgments:**We thank all contributors to the open-source programs “pair interaction” and “ARC” as well as R. Côté, W. D. Phillips, N. Šibalić, and J. Deiglmayr for valuable discussions.

**Funding:**We acknowledge support by the DNRF through a Niels Bohr Professorship for T.P. and funding by MPG. This project has received funding from the European Union’s Horizon 2020 research and innovation program under grant agreement 817482 (PASQuanS) and the European Research Council (ERC) 678580 (RyD-QMB) and also from project 319278 (UQUAM) of the Seventh Framework Programme. We also acknowledge funding from Deutsche Forschungsgemeinschaft (project BL 574/15-1) within SPP 1929 (GiRyd).

**Author contributions:**S.H. and J.R. acquired the data underlying this study and, together with J.Z. and A.R.-A., they maintained and improved the experimental setup. V.W. and T.P. contributed the theoretical predictions. I.B. and C.G. supervised the study. All authors worked on the interpretation of the data and contributed to the final manuscript.

**Competing interests:**The authors declare no competing interests.

**Data and materials availability:**The data that support the plots presented in this paper are publicly available from the Open Access Data Repository of the Max Planck Society (

*36*).