  Open quantum many-body systems with controllable dissipation can exhibit
novel features in their dynamics and steady states. A paradigmatic example is
the dissipative transverse field Ising model. It has been shown recently that
the steady state of this model with all-to-all interactions is genuinely
non-equilibrium near criticality, exhibiting a modified time-reversal symmetry
and violating the fluctuation-dissipation theorem. Experimental study of such
non-equilibrium steady-state phase transitions is however lacking. Here we
propose realistic experimental setups and measurement schemes for current
trapped-ion quantum simulators to demonstrate this phase transition, where
controllable dissipation is engineered via a continuous weak optical pumping
laser. With extensive numerical calculations, we show that strong signatures of
this dissipative phase transition and its non-equilibrium properties can be
observed with a small system size across a wide range of system parameters. In
addition, we show that the same signatures can also be seen if the dissipation
is instead achieved via Floquet dynamics with periodic and probabilistic
resetting of the spins. Dissipation engineered in this way may allow the
simulation of more general types of driven-dissipative systems or facilitate
the dissipative preparation of useful many-body entangled states.
