**Figure 5.** (a) Three potential wells in the *x*–*y* plane corresponding to three different eigenstates of *H*_{int}. The solid red, black dashed and blue dotted lines correspond to |ψ_{1}〉, |ψ_{2}〉 and |ψ_{3}〉, respectively. The black dot indicates the initial position of the system for the dynamics discussed in figure 6 and the arrow indicates the direction of motion. (b) Imaginary part of \tilde{A}_{12}^{(1)}=[\tilde{A}_{21}^{(1)}]^* for = 0. (c) Real parts of \tilde{A}_{11}^{(2)} (solid red line), \tilde{A}_{22}^{(2)} (dashed black line) and \tilde{A}_{12}^{(2)}=[\tilde{A}_{21}^{(2)}]^* (dotted blue line) for = 0. (d) Matrix elements of the commutator *C* in equation (34). The red solid line shows *C*_{11} = −*C*_{22} and the black dashed line represents *C*_{12} = *C*_{21}. In (a)–(d), we set Δ = −1.13|δ|. All components of \boldsymbol{\tilde{A}} that are not shown in (b) and (c) are zero.

**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.