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Quantum tunneling is an important dynamical process in ultracold many-body systems subjected to complex trapping environments such as optical lattices. The competition of the tunable interactions and the geometry of the external trap leads to a plethora of different tunneling scenarios. Paradigm examples are the self-trapping and correlated pair tunneling of repulsively interacting bosons and, for stronger interactions, to independent fermion tunneling in a single dimension. We use exact ab initio methods, such as the Multi-Configuration Time-Dependent Hartree Method, to study novel tunneling scenarios and mechanisms for pure and mixed bosonic systems. Windows of enhanced tunneling in a regime of general suppression of tunneling can be opened in the strong interaction regime by higher band contributions in multi-well traps. The resulting process of interband tunneling is identified and shown to comprise single-boson tunneling, two-boson correlated tunneling and conditional tunneling processes in general. For mixtures one species creates an effective potential for the second species, and an induced attraction leading to attractive pair tunneling is found. The latter leads to fermionic pair dynamics in the strong interaction regime. Interaction inhomogeneities in the regime of strong correlations are shown to create a plethora of novel tunneling resonances.
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