  We study the non-equilibrium dynamics of two dimensional planar ion Coulomb
crystals undergoing a structural buckling transition to a three plane
configuration, driven by a reduction of the transverse confining frequency.
This phase transition can be theoretically modeled using a mapping to a two
dimensional Ginzburg-Landau theory with complex order parameter field. We
demonstrate that finite rate quenches result in creation of stable topological
vortices, which are localized point regions around which the phase of the order
parameter field winds a multiple of 2{\pi}. The density of the defects as a
function of quench rate is investigated using molecular dynamics simulations
and its scaling is shown to be consistent with Kibble-Zurek theory of defect
formation. Following the quench, the annihilation of vortex and anti-vortex
pairs results in the relaxation of defect density that follows a diffusive
scaling with a logarithmic correction. This work highlights the potential for
investigating complex non-equilibrium statistical physics of topological
defects in an experimentally accessible ion trap setting.
