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One of the challenges with quantum simulation in ion traps is that theeffective spin-spin exchange couplings are not uniform across the lattice. Thiscan be particularly important in Penning trap realizations where the presenceof an ellipsoidal boundary at the edge of the trap leads to dislocations in thecrystal. By adding an additional anharmonic potential to better controlinterion spacing, and a triangular shaped rotating wall potential to reduce theappearance of dislocations, one can achieve better uniformity of the ionicpositions. In this work, we calculate the axial phonon frequencies and thespin-spin interactions driven by a spin-dependent optical dipole force, anddiscuss what effects the more uniform ion spacing has on the spin simulationproperties of Penning trap quantum simulators. Indeed, we find the spin-spininteractions behave more like a power law for a wide range of parameters.

Theoretical basis for quantum simulation with a planar ionic crystal in a Penning trap using a triangular rotating wall