Quantum motion of two trapped ions in one dimension

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Abstract
We develop models for the motion of two trapped ions in one dimension, including both harmonic trapping forces and the Coulomb interaction between the ions. Mathematical and physical consistency dictates that the divergence of the Coulomb interaction at small distances be tempered with a soft core. A split-operator method is used to analyze numerically the energies and eigenstates of the motion of the two ions. Classification and closed-form expressions are also presented for the energies and the eigenstates in the limits when the equilibrium distance due to the Coulomb repulsion is either much smaller or much larger than the amplitude of the zero-point motion of the ions. Implications of the results for the properties of two identical ions with Bose-Einstein or Fermi-Dirac statistics are discussed.

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