  Our work analyzes the potential of ion traps for the experimental simulation
of non-equilibrium phase transitions observed in certain spin-chain models
which can be mapped to free-fermion systems. In order to make the dynamics more
accessible to an experimenter, we first consider relatively small systems, with
few particles. We analyze phase transitions in the non-equilibrium asymptotic
regimes of an XY spin chain with a transverse magnetic field and coupled to
Markovian baths at the end sites. We study a static open system and a case when
the spin chain is periodically kicked. Notably, in the latter case for some
anisotropy parameters the dependence on the system size converges rapidly to
the many-particle limit, thus facilitating the experimental observation of the
dynamics. We also define local observables that indicate the presence of the
quantum phase transitions of interest, and we study the effects of the
long-range character of the typical interactions obtained in ion traps.
