  Rapid separation of linear crystals of trapped ions into different subsets is
critical for realizing trapped ion quantum computing architectures where ions
are rearranged in trap arrays to achieve all-to-all connectivity between
qubits. We introduce a general theoretical framework that can be used to
describe the separation of same-species and mixed-species crystals into smaller
subsets. The framework relies on an efficient description of the evolution of
Gaussian motional states under quadratic Hamiltonians that only requires a
special solution of the classical equations of motion of the ions to describe
their quantum evolution under the influence of a time-dependent applied
potential and the ions' mutual Coulomb repulsion. We provide time-dependent
applied potentials suitable for separation of a mixed species three-ion crystal
on timescales similar to that of free expansion driven by Coulomb repulsion,
with all modes along the crystal axis starting and ending close to their ground
states. Three separately-confined mixed species ions can be combined into a
crystal held in a single well without energy gain by time-reversal of this
separation process.
