Spin-motion coupling in a circular Rydberg state quantum simulator: case of two atoms

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.