Abstract: Non-equilibrium quantum many-body systems can display fascinating phenomena relevant for various fields in science ranging from physics, to chemistry, and ultimately, for the broadest possible scope, life itself. The challenge with these systems, however, is that the powerful formalism of statistical physics, which have allowed a classification of quantum phases of matter at equilibrium does not apply. Therefore, using controllable cold atomic systems to shed light on the organizing principles and universal behaviors of dynamical quantum matter is highly appealing. One emerging paradigm is the dynamical phase transition (DPT) characterized by the existence of a long-time-average order parameter that distinguishes two non-equilibrium phases. I will report the observation of a DPT in two different but complementary systems: a trapped quantum degenerate Fermi gas  and long lived arrays of atoms in an optical cavity . I will show how these systems can be used to simulate iconic models of quantum magnetism with tunable parameters and to probe the dependence of their associated dynamical phases on a broad parameter space. Besides advancing quantum simulation our studies pave the ground for the generation of metrologically useful entangled states which can enable real metrological gains via quantum enhancement.
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