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Laser cooled ions trapped in a linear rf-Paul trap are ideal candidates for carrying out quantum simulation of many-body systems. Such a collection of trapped ions exchange vibrational quanta through the Coulomb interaction. In this talk, we will investigate alternative methods for thermally induced nonclassicality generation in two very different contexts. First, in the harmonic approximation, the linear interaction can be shown to be anharmonically modulated by an elementary coupling to the internal two-level structure of the ions. Driven by thermal energy in the passively coupled oscillators, the nonlinear interaction autonomously and unconditionally generates entanglement between the mechanical modes of the ions. In addition, we demonstrate multiqubit enhancement of such thermally induced entanglement. In the second part of the talk, we will go beyond the harmonic approximation which leads to the realization of nonlinear Hamiltonians, such as both the degenerate and nondegenerate parametric oscillator Hamiltonians. Stimulated by small thermal energy, the strong degenerate trilinear coupling generates large amounts of nonclassicality which is detectable by more than 3~dB of distillable quadrature squeezing, in contrast to the nondegenerate trilinear coupling. Substantial entanglement can be generated in tandem with the nonclassicality generation via linear coupling to a third mode, achieving the entanglement potential of nonclassical states. Such entanglement, although not captured by the covariance matrix, grows with the mean number of split thermal quanta. We extend our investigation to higher-order degenerate quanta splitting processes to demonstrate that this non-Gaussian entanglement is already common for strongly nonlinear systems at the low-energy thermal level. |