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Linear arrays of trapped and laser cooled atomic ions are a versatile platform for studying emergent phenomena in strongly-interacting many-body systems. Effective spins are encoded in long-lived electronic levels of each ion and made to interact through laser mediated optical dipole forces. The advantages of experiments with cold trapped ions, including high spatiotemporal resolution, decoupling from the external environment, and control over the system Hamiltonian, are used to measure quantum effects not always accessible in natural condensed matter samples. In this review we highlight recent work using trapped ions to explore a variety of non-ergodic phenomena in long-range interacting spin-models which are heralded by memory of out-of-equilibrium initial conditions. We observe long-lived memory in static magnetizations for quenched many-body localization and prethermalization, while memory is preserved in the periodic oscillations of a driven discrete time crystal state.
We introduce an experimentally accessible network representation for many-body quantum states based on entanglement between all pairs of its constituents. We illustrate the power of this representation by applying it to a paradigmatic spin chain mode
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
We analyze a crossover between ergodic and non-ergodic regimes in an interacting spin chain with a dilute density of impurities, defined as spins with a strong local potential. The dilute limit allows us to greatly suppress finite size effects and un
Symmetries play a central role in single-particle localization. Recent research focused on many-body localized (MBL) systems, characterized by new kind of integrability, and by the area-law entanglement of eigenstates. We investigate the effect of a
We study measures of decoherence and thermalization of a quantum system $S$ in the presence of a quantum environment (bath) $E$. The entirety $S$$+$$E$ is prepared in a canonical thermal state at a finite temperature, that is the entirety is in a ste