Long-range interactions between rubidium and potassium Rydberg atoms

We investigate the long-range, two-body interactions between rubidium and potassium atoms in highly excited $(n=70)$ Rydberg states. After establishing properly symmetrized asymptotic basis states, we diagonalize an interaction Hamiltonian consisting of the standard Coulombic potential expansion and atomic fine structure to calculate electronic potential energy curves. We find that when both atoms are excited to either the $70s$ state or the $70p$ state, both the $\mathrm{\Omega }=0^{+}$ symmetry interactions and the $\mathrm{\Omega }=0^{-}$ symmetry interactions demonstrate a deep potential well capable of supporting many bound levels; the sizes of the corresponding dimer states are of the order of 2.25 $\mu \mathrm{m}$. We establish $n$-scaling relations for the equilibrium separation $R_e$ and the dissociation energy $D_e$ and find these relations to be consistent with similar calculations involving the homonuclear interactions between rubidium and cesium. We discuss the specific effects of $\ell$ mixing and the exact composition of the calculated potential well via the expansion coefficients of the asymptotic basis states. Finally, we apply a Landau-Zener treatment to show that the dimer states are stable with respect to predissociation.

Figure 1

Two Rydberg atoms well separated from each other; each consists of a $+1$ nuclear core and a single, highly excited valence electron $e_i$. Here, the nuclear distance $R$ is greater than the LeRoy radius (see text) and thus much larger than either electron's distance $r_i$ from its respective nuclear core.

Figure 2

Long-range potential energy curves corresponding to the interactions between one rubidium atom and one potassium atom excited to the same state. (a) Curves correspond to the $\mathrm{\Omega }=0^{+}$ symmetry with both atoms excited to the $70s$ state, (b) curves correspond to the $\mathrm{\Omega }=0^{-}$ symmetry with both atoms excited to the $70s$ state, (c) curves correspond to the $\mathrm{\Omega }=0^{+}$ symmetry with both atoms excited to the $70p$ state, and (d) curves correspond to the $\mathrm{\Omega }=0^{-}$ symmetry with both atoms excited to the $70p$ state. We highlight the potential energy wells in red and label their respective asymptotic states.

Figure 3

(a) Long-range potential energy curves corresponding to the $\mathrm{\Omega }=0^{+}$ symmetry interactions of one potassium Rydberg atom and one rubidium Rydberg atom; both atoms have been excited to the $70s$ state. We note the existence of an $\sim 600$-MHz-deep well associated with the ${69}^{\left(\mathrm{K}\right)}s+{69}^{\left(\mathrm{Rb}\right)}{d}_{5/2}$ asymptotic level, capable of supporting many bound states. We explicitly label the equilibrium separation $R_e$ and the dissociation energy $D_e$. Inset: Zoom-in on the deepest part of the well, showing the first few bound vibrational levels; corresponding energies and classical turning points are listed in Table 4. (b) Scaling relations for the equilibrium separation $R_e$ vs the principal quantum number $n$: $R_e\sim n^{12/5}$. (c) Scaling relations for the dissociation energy $D_e$ vs the principal quantum number $n$. Three results (see text) are shown: $D_e\sim n^{-4}$ (blue line) gives a poor agreement, $D_e\sim n^{-3}$ (dashed black line) gives a good agreement, and $D_e\sim n^{-3}+n^{-4}$ (red line) gives the best agreement.

Figure 4

(a) $\mathrm{\Omega }=0^{+}$ potential energy curves describing the interactions between one potassium Rydberg atom and one rubidium Rydberg atom: both atoms have been excited to the $70s$ state. We highlight curves corresponding to the electronic states that contribute the most to the formation of the well (see text). Inset: Zoom-in on the region near the avoided crossing, indicating the relevant features of the Landau-Zener treatment: $\mathrm{\Delta }$ is the energy gap between the adiabatic energies ${\epsilon }_1$ and ${\epsilon }_2$ at closest approach; these adiabatic energies correspond to the two adiabatic states $|{\chi }_1\rangle$ and $|{\chi }_2\rangle$ at the avoided crossing. (b) Probabilities ${\left|c_j\left(R\right)\right|}^2$ of the electronic basis states $|j\rangle$ most responsible for the formation of the well (see text). After $\sim 60000a_0$, the $\ell$ mixing is negligible: the probability coefficient corresponding to ${69}^{\left(\mathrm{K}\right)}s{+}^{\left(\mathrm{Rb}\right)}70d_{5/2}$ approaches 1 (see dashed line) and all other probability coefficients decay to 0. Inset: Zoom-in on the region near the avoided crossing. All panels use the same labeling and/or color scheme.