Ergodicity and chaos play an integral role in the dynamical behavior of many-particle systems and are crucial to the formulation of statistical mechanics. Still, a general understanding of how randomness and chaos emerge in the dynamical evolution of closed quantum systems remains elusive. Here, we develop an experimental platform for the realization of canonical quantum chaotic Hamiltonians based on quantum simulation with synthetic lattices. We map the angular momentum projection states of an effective quantum spin onto the linear momentum states of a $^{87}$Rb Bose-Einstein condensate, which can alternatively be viewed as lattice sites in a synthetic dimension. This synthetic lattice, with local and dynamical control of tight-binding lattice parameters, enables new capabilities related to the experimental study of quantum chaos. In particular, the capabilities of our system let us tune the effective size of our spin, allowing us to illustrate how classical chaos can emerge from a discrete quantum system. Moreover, spectroscopic control over our synthetic lattice allows us to explore unique aspects of our spins dynamics by measuring the out-of-time-ordered correlation function, and enables future investigations into entirely new classes of chaotic systems.
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