Systems far from equilibrium are of fundamental scientific interest, as they can exhibit properties that differ dramatically from those of the same system at equilibrium. Certain classes of out-of-equilibrium systems can even sustain long-lived states with emergent material properties. This talk presents the realization of quantum states out of equilibrium in two different settings. First, we discuss the emergence of self-stabilized square lattice patterns in a superfluid with periodically modulated interactions. We show that these patterned states share key physical properties to a seemingly different equilibrium system, namely supersolids. To demonstrate this correspondence, we probe the excitation spectrum of such a patterned state, identifying two distinct sound modes associated with spontaneously broken U(1) and translational symmetries. Second, we present the non-equilibrium preparation of a U(1) quantum-spin-liquid state in a cesium quantum gas microscope. We implement a two-dimensional monomer-dimer model governed by a U(1) gauge theory. Starting from an initial product state of monomers, we dynamically generate lakes of Rokhsar-Kivelson type U(1) dimer spin-liquids, and study the state’s coherence using interferometric round-trip protocols.