This thesis presents a theoretical study of Bose-Einstein condensation (BEC) and Bardeen-Cooper-Schrieffer (BCS) pairing states in inhomogeneous systems of cold atoms and of electrons. Features of spatially separated phases are explored, with particular focus on the behavior of the condensed phase and its experimental measures. Three specific systems are addressed below.First, we study bosonic atoms in three-dimensional optical lattices in the presence of an external spherical harmonic trapping potential. We investigate the critical value associated with the lattice depth and interaction strength below which the system undergoes a quantum phase transition from a global BEC phase to a coexistence of local BEC and Mott-insulating phases. We discuss the ground state properties, excitations, and experimental signatures of the condensate surrounded by the Mott-insulators.BCS pairing in fermionic atoms of two spin species that are confined to spatially separated trapping potentials is investigated next. We investigate the one-dimensional limit and find that, with increasing separation between the spin-dependent traps, the fermions undergo a transition from a global fully-paired phase to a coexistence of a fully-paired phase, a spin-imbalanced phase with oscillatory pairing, the so-called Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state, and an unpaired completely spin-polarized phase. We present numerical profiles of key parameters of the phase diagram as well as observable signatures of the oscillatory pairing phase.The third topic is that of transport physics in a superconductor-ferromagnetic-metal (S/F) hybrid in which superconducting phases and ferromagnetic normal phases are artificially combined. We model the interface between the S and F regions and discuss possible scattering processes at the interface. We apply the Blonder-Tinkham-Klapwijk treatment with the interfacial model to calculate resistance of the system. These results explain recent experimental observations.
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