Identify a second Rydberg state that yields a negative and dominant cross van der Waals interaction

Determine a specific even-parity Rydberg state |\tilde{r}\rangle to weakly couple with the primary Rydberg state |r\rangle (with C_6 > 0 for the |r r\rangle van der Waals interaction) such that the cross-state van der Waals coefficient between |r\rangle and |\tilde{r}\rangle satisfies \tilde{C}_6 < 0 and |\tilde{C}_6| \gg C_6, thereby ensuring the effective interaction parameters satisfy F_6 < 0 with |F_6| \approx C_6. This identification is required to realize nearest- and next-nearest-neighbor spin interactions of opposite sign in the proposed mixed-field Ising model implemented with Rydberg atoms.

Background

The paper proposes realizing a mixed-field spin-1/2 Ising model with sign-inverted next-nearest-neighbor interaction using Rydberg atoms, by weakly coupling a Rydberg state |r\rangle to another Rydberg state |\tilde{r}\rangle. The effective interaction between two atoms in the Rydberg-dressed Rydberg (RDR) state includes both the bare van der Waals term for |r r\rangle and a second-order perturbative contribution parameterized by F_6, allowing control of the interaction sign.

Achieving a sign inversion between nearest- and next-nearest-neighbor couplings requires the cross van der Waals coefficient \tilde{C}6 between |r\rangle and |\tilde{r}\rangle to be negative and large in magnitude compared to C_6. While the authors outline an experimental strategy and give examples (e.g., for ^{87}Rb choosing |r\rangle as nS{1/2} and considering |\tilde{r}\rangle in \tilde{n}D_{3/2}), they explicitly note that the specific Rydberg state |\tilde{r}\rangle meeting these quantitative conditions is not presently known and suggest measuring \tilde{C}_6 experimentally as a potential route to finding a suitable state.