Dissipative Floquet Dynamics: from Steady State to Measurement Induced Criticality in Trapped-ion Chains*

Quantum systems evolving unitarily and subject to quantum measurements exhibit various types

of non-equilibrium phase transitions, arising from the competition between unitary evolution and

measurements. Dissipative phase transitions in steady states of time-independent Liouvillians and

measurement induced phase transitions at the level of quantum trajectories are two primary ex-

amples of such transitions. Investigating a many-body spin system subject to periodic resetting

measurements, we argue that many-body dissipative Floquet dynamics provides a natural frame-

work to analyze both types of transitions. We show that a dissipative phase transition between a

ferromagnetic ordered phase and a paramagnetic disordered phase emerges for long-range systems

as a function of measurement probabilities. A measurement induced transition of the entanglement

entropy between volume law scaling and area law scaling is also present, and is distinct from the

ordering transition. The ferromagnetic phase is lost for short range interactions, while the volume

law phase of the entanglement is enhanced. An analysis of multifractal properties of wave func-

tion in Hilbert space provides a common perspective on both types of transitions in the system.

Our findings are immediately relevant to trapped ion experiments, for which we detail a blueprint

proposal based on currently available platforms.