  Rydberg atoms are remarkable tools for the quantum simulation of spin arrays.
Circular Rydberg atoms open the way to simulations over very long time scales,
using a combination of laser trapping of the atoms and spontaneous-emission
inhibition, as shown in the proposal of a XXZ spin-array simulator based on
chains of trapped circular atoms [T.L. Nguyen $\textit{et al.}$, Phys. Rev. X
8, 011032 (2018)]. Such simulators could reach regimes (thermalization, glassy
dynamics) that are out of the reach of those based on ordinary,
low-angular-momentum short-lived Rydberg atoms. Over the promised long time
scales, the unavoidable coupling of the spin dynamics with the atomic motion in
the traps may play an important role. We study here the interplay between the
spin exchange and motional dynamics in the simple case of two interacting
circular Rydberg atoms confined in harmonic traps. The time evolution is solved
exactly when the position dependence of the dipole-dipole interaction terms can
be linearized over the extension of the atomic motion. We present numerical
simulations in more complex cases, using the realistic parameters of the
simulator proposal. We discuss three applications. First, we show that
realistic experimental parameters lead to a regime in which atomic and spin
dynamics become fully entangled, generating interesting non-classical motional
states. We also show that, in other parameter regions, the spin dynamics
notably depends on the initial temperature of the atoms in the trap, providing
a sensitive motional thermometry method. Last, and most importantly, we discuss
the range of parameters in which the motion has negligible influence over the
spin dynamics.
