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Abstract: Once cooled to temperatures as low as one nano-Kelvin, the quantum mechanical wave nature of cold atoms becomes apparent. By controlling the interaction between these atoms, a dilute gas of merely ten thousand atoms—about a million times less dense than air—can exhibit novel nonlinear wave phenomena and fascinating self-patterning dynamics, some of which may be difficult to observe in other nonlinear systems. In this talk, I will discuss the preparation, control, and measurement of quantum matter waves trapped in an optical box. I will demonstrate how triggering instabilities in these nonlinear matter waves can induce self-patterning dynamics, forming spatially or temporally ordered structures and density defects such as vortex dipoles and soliton lasers. As quantum fluctuations in the matter waves primarily seed these pattern-forming instabilities, I will explore how quantum entanglement may arise from the underlying dynamics. One concrete example is the quasiparticle pair production and the formation of Einstein–Podolsky–Rosen pairs of entangled phonons within a matter wave.