  We investigate the back-flow of information in a system with a second-order
structural phase transition, namely a quasi one-dimensional Coulomb crystal.
Using standard Ramsey interferometry which couples a target ion (the system) to
the rest of the chain (a phononic environment), we study the non-Markovian
character of the resulting open system dynamics. We study two different
time-scales and show that the back-flow of information pinpoints both the phase
transition and different dynamical features of the chain as it approaches
criticality. We also establish an exact link between the back-flow of
information and the Ramsey fringe visibility.
