Universal relations in ultracold polarized Fermi gases

Ultracold quantum gases are an ideal toolbox for simulating complex condensed or nuclear matter systems and to investigate fundamental quantum properties of matter. In this thesis, we will investigate universal properties connecting the high-momentum tail of the momentum distribution, the short-range correlations and the thermodynamics, encapsulated in Tan's relations. These relations are especially useful in the strongly interacting case, where perturbative approaches usually fail. With the aid of Tan's universal relations, we can still come to general conclusions about strongly interacting quantum gases. In particular, the momentum distribution exhibits a characteristic algebraic decay, unlike the exponential decay of the non-interacting case. The main focus in this thesis is on the one-dimensional, fermionic case, where we study the highly polarized case (the one-dimensional Fermi polaron), verifying Tan's relations using a variety of theoretical tools. In addition, we show that localized systems exhibit a universal, dynamical instability to delocalization when a short-range interaction between particles is switched off rapidly. This delocalization process relies on the algebraic decay of the momentum distribution, which guarantees that at least some of the delocalized single-particle states are occupied with a finite probability. Finally, we investigate the Bardeen-Cooper-Schrieffer (BCS) to Bose-Einstein condensation (BEC) crossover for the three-dimensional Fermi gas and develop a novel method to describe the breakdown of the Fermi liquid description in the vicinity of the critical temperature for superfluidity, in good agreement with a recent experiment.