## (a) Lowest vibrational states in the potential shown in figure 3(a) for Δ = −3|δ| and Ω_{L}/|δ| = 2.8 **×** 10^{−6}

**Figure 4.** (a) Lowest vibrational states in the potential shown in figure 3(a) for Δ = −3|δ| and Ω_{L}/|δ| = 2.8 **×** 10^{−6}. The energy difference of the vibrational states is ω_{vib} ≈ 0.015|δ|. (b) Lowest vibrational state for different quantum numbers \mathcal {M} and on an energy scale defined by Ω_{L}.

**Abstract**

We show that the dipole–dipole interaction between two Rydberg atoms can lead to substantial Abelian and non-Abelian gauge fields acting on the relative motion of the two atoms. We demonstrate how the gauge fields can be evaluated by numerical techniques. In the case of adiabatic motion in a single internal state, we show that the gauge fields give rise to a magnetic field that results in a Zeeman splitting of the rotational states. In particular, the ground state of a molecular potential well is given by the first excited rotational state. We find that our system realizes a synthetic spin–orbit coupling where the relative atomic motion couples to two internal two-atom states. The associated gauge fields are non-Abelian.