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Intrinsic anharmonic effects on the phonon frequencies and effective spin-spin interactions in a quantum simulator made from trapped ions in a linear Paul trap

The Coulomb repulsion between ions in a linear Paul trap give rise to anharmonic terms in the potential energy when expanded about the equilibrium positions. We examine the effect of these anharmonic terms on the accuracy of a quantum simulator made from trapped ions. To be concrete, we consider a linear chain of $\text{Yb}^{171+}$ ions stabilized close to the zigzag transition. We find that for typical experimental temperatures, frequencies change by no more than a factor of $0.01\%$ due to the anharmonic couplings. Furthermore, shifts in the effective spin-spin interactions (driven by a spin-dependent optical dipole force) also tend to be small for detunings to the blue of the transverse center-of-mass frequency. However, detuning the spin interactions near other frequencies can lead to nonnegligible anharmonic contributions to the effective spin-spin interactions. We also examine an odd behavior exhibited by the harmonic spin-spin interactions for a range of intermediate detunings, where nearest neighbor spins with a larger spatial separation on the ion chain interact more strongly than nearest neighbors with a smaller spatial separation.