Ultracold atomic gases have emerged as an ideal laboratory system to emulate many-body physics in an unprecedentedly controllable manner. Numerous many-body quantum states and phases have been experimentally explored and characterized using the ultracold atomic gases, offering new insights into many exciting physics ranging from condensed matters to cosmology. In this thesis, we will present a systematic numerical study of a novel experimental system, population imbalanced two-component ultracold Fermi gases. We explore the phase diagram of this system in both 3D and 1D especially focusing on the exotic Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) phase, which is characterized by a spatially oscillating order parameter. In 3D, we solve for the stationary states of trapped imbalanced Fermi gases in a wide range of parameter space with a home-made parallel eigen-solver for Bogoliubov-de Gennes (BdG) equations. Our results show that there exists a metastable state with a FFLO type oscillating order parameter. In 1D, we simulate the dynamical expansion of the population imbalanced Fermi gases from the trap. A numerically quasi-exact scheme, time-evolving block decimation (TEBD), is introduced for the comparative studies with the solution of the time-dependent BdG equation. Our results predict that the existence of FFLO states will leave conspicuous signatures in the density profiles during the expansion. For further understanding of the interplay between the population imbalance and two-body pairing interaction between two spin components, we also study the spin transport properties through trapped ultracold Fermi gases. The preliminary results will be discussed.

U6Fe has the highest superconducting transition temperature TSC ~ 4 K out of all of the U-based compounds. Unusually, the Pauli limit (1:84TSC = 7:36 T) is less than the observed critical field for both the a and c axes in this tetragonal material. Neither Pauli or usual BCS orbital limit is apparently respected. In order to explain why superconductivity exceeds the Pauli limit, it must be considered that either the superconducting state is unaffected by paramagnetic effects, or there is a large amount of spin-orbit scattering. Superconductivity is in the dirty limit for typical samples of U6Fe, which means that the latter cannot be precluded. Another unusual property of the superconducting state of U6Fe is that TSC has a positive dependence on the applied pressure P, for P < 4 kbar. This combined with other subtle signals in various measurements have led to the suggestion that a Charge Density Wave (CDW) state may exist in U6Fe below 110 K. The CDW state is typically favoured by materials with low-dimensional structural features such as chains of atoms. Such materials, if superconductors, are also candidates to exhibit the sought-after Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state, an unusual state in which the superconducting order parameter is modulated in real space. The FFLO is expected to be suppressed except in clean, Pauli limited materials. Therefore if U6Fe can be produced at high enough quality to bring the superconducting state into the clean limit, it would be a potential candidate for an FFLO state. Part of this project discusses apparatus and techniques applied with the goal of producing such quality samples of U6Fe. We have succeeded in the application of the Solid State Electrotransport (SSE) method to purifying samples, and have been able to replicate the highest Residual Resistivity Ratios (RRRs) achieved (~9, compared to 4 for typical samples), but for single crystals instead of the polycrystals produced in the past. In parallel with the progress made towards higher quality samples of U6Fe, a new X-ray scanner has been developed for grain mapping of samples. This has found application in the course of our synthesis studies. The best quality samples have been studied by X-ray diffraction on the XMaS beamline at the ESRF in Grenoble, France. Below TCDW ~ 10 K, satellites at (δH; δK; 0) = (±0:11;±0:11; 0) were observed that confirm a CDW state, albeit at much lower temperatures than anticipated. By examination of systematic satellite absences we have determined that the displacement vector → u is perpendicular to the modulation direction in k-space. Additionally it has been found that the symmetry of the lattice below TCDW is reduced from that of the room temperature I4=mcm structure. The appearance of additional Bragg peaks below ~110 K during these experiments were later cast into doubt by multiple scattering. We have, however, detected a signal in the form of a jump at ~110 K in specific heat measurements of our samples. These measurements also show a kink near to TCDW. We have additionally extended the investigation of the effect of pressure on the superconducting state. The maximum of TSC is confirmed in our samples, and the subsequent suppression of TSC and Hc2 is investigated up to 8 GPa. We have analysed our Hc2(T) curves at different pressures under a simple two-band model that fits the observed trends well and suggest that at the highest pressures U6Fe is approaching even more unusually enhanced Hc2 values.