# Semi-classical simulation of the system dynamics in the *x*–*y* plane

**Figure 6.** Semi-classical simulation of the system dynamics in the *x*–*y* plane. The initial conditions are \langle \boldsymbol{\rho } \rangle = 1.5 R_0 and \langle \boldsymbol{p} \rangle =\boldsymbol{0}. The system is prepared in the internal state |ψ_{2}〉 at *t* = 0. The parameters are Δ = −1.13|δ| and Ω_{L}/|δ| = 2.8 **×** 10^{−6}, corresponding to |δ| = 2π **×** 11.4 MHz, ^{39}K atoms with principal quantum number *n* = 30 and *R*_{0} = 2.85 μm. We neglect effects due to the finite lifetime of the molecule (*t*|δ| ≈ 1300) and thus restrict the analysis to times that are significantly smaller. (a) Position 〈ρ〉 as a function of time. (b) Population of the internal states. The red solid line corresponds to |ψ_{2}〉 and the blue dashed line shows the population in |ψ_{1}〉. The black dotted line shows the sum of the population in |ψ_{1}〉 and |ψ_{2}〉.

**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.