Effective Three-Body Interactions in $\mathrm{Cs}(6s)\text{−}\mathrm{Cs}(nd)$ Rydberg Trimers

Ultralong-range Rydberg trimer molecules are spectroscopically observed in an ultracold gas of $\mathrm{Cs}(n{d}_{3/2})$ atoms. The anisotropy of the atomic Rydberg state allows for the formation of angular trimers, whose energies may not be obtained from integer multiples of dimer energies. These nonadditive trimers coexist with Rydberg dimers. The existence of such effective three-body interactions is confirmed with the observation of asymmetric line profiles and interpreted by a theoretical approach that includes relativistic spin interactions. Simulations of the observed spectra with and without angular trimer lines lend convincing support to the existence of effective three-body interactions.

Figure 1

The angular trimer potential energy curves (a) for $R_1=R_2=1868$ $a_0$ for excitation into the $\mathrm{Cs}(34d_{3/2})$ Rydberg state, without any spin-dependent terms in the interaction Hamiltonian, i.e., $E_{\pm }$ terms in Eq. (2) rescaled by $E_{\dim }(R=1868a_0)$. The primitive orbitals (signs indicated) whose superpositions ${\psi }_{\pm }(\mathit{r};{\mathit{R}}_1;{\mathit{R}}_2)$ produce the eigenvalues $E_{\pm }$, are illustrated in (b). The angular configuration of the Rydberg core (at the origin) and the two ground state atoms (unit vectors ${\widehat{R}}_1$, ${\widehat{R}}_2$) are represented by the central sphere and the outer spheres, respectively. In (c) the relativistic spin-dependent interactions deform the curves $E_{\pm }$ and lift degeneracies, leading to modified angular potentials. Here $E_{\dim }^{\text{triplet}}=71\mathrm{MHz}$ ($E_{\dim }^{\text{mixed}}=34\mathrm{MHz}$) corresponds to the energy at the minimum of the outermost well of the triplet dominated (mixed singlet or triplet) dimer PES; see the dashed blue (red) curve in Fig. 2.

Figure 2

Comparison between the experimental spectrum (black line) and theoretical results for molecules close to the $\mathrm{Cs}(34d_{3/2})$ $F_1=F_2=3$ dissociation energy. Vibrational wave functions (colored full lines) are presented for a few diatomic potential curves (dashed lines) that have been selected from all potential energy curves (gray background). The offset of the wave function corresponds to its vibrational energy. The lengths of the colored bars on top of the experimental spectrum are the theoretical line strengths. The computed trimer histogram (bin width 700 kHz) is superimposed in orange. Dimer states localized in the most outer wells are marked with a star.

Figure 3

Comparison of the experimental spectra (solid blue lines) and theoretical simulations including molecular dimer and trimer signals (dashed dotted black lines) close to the $\mathrm{Cs}(34d_{3/2}\mathrm{and}36d_{3/2})$ atomic Rydberg lines with $F_1=F_2=3$. To estimate the impact of trimer signals we also present theoretical simulations including only dimer states (dashed gray lines). The red circles highlight the energy regions where the contributions of the nonadditive trimers are most prominent in signal height and energy spread. The asymmetric long shoulders of these peaks can be explained only by the nonadditive trimer signals. The upper panel displays a magnification of the region with dominant trimer contributions for $34d_{3/2}$.