Abstract:
A superfluid-insulator transition is known to occur in strongly disordered Fermi gases, in both the BCS and BEC regimes; here, we address the properties of this transition across the BEC-BCS crossover. We argue that the critical disorder strength at which superfluidity is lost changes non-monotonically with detuning from Feshbach resonance, and that a reentrant superfluid phase arises for detunings near the fermionic mobility edge. Our analysis of the intermediate regime is quantitatively valid for narrow resonances and near four dimensions, and provides a simple physical picture of this regime, in terms of two distinct but coexisting insulators.

Abstract:
We explore the zero temperature phase behavior of a two-dimensional two-component atomic Fermi gas with population and mass imbalance in the regime of the BEC-BCS crossover. Working in the mean-field approximation, we show that the normal and homogeneous balanced superfluid phases are separated by an inhomogeneous superfluid phase of Fulde-Ferrel-Larkin-Ovchinnikov (FFLO) type. We obtain an analytical expression for the line of continuous transitions separating the normal and inhomogeneous FFLO phases. We further show that the transition from the FFLO phase to the homogeneous balanced superfluid is discontinuous leading to phase separation. If the species have different masses, the superfluid phase is favored when the lighter species is in excess. We explore the implications of these findings for the properties of the two-component Fermi gas in the atomic trap geometry. Finally, we compare and contrast our findings with the predicted phase behavior of the electron-hole bilayer system.

Abstract:
Multiply quantized vortices in the BCS-to-BEC evolution of p-wave resonant Fermi gases are investigated theoretically. The vortex structure and the low-energy quasiparticle states are discussed, based on the self-consistent calculations of the Bogoliubov-de Gennes and gap equations. We reveal the direct relation between the macroscopic structure of vortices, such as particle densities, and the low-lying quasiparticle state. In addition, the net angular momentum for multiply quantized vortices with a vorticity $\kappa$ is found to be expressed by a simple equation, which reflects the chirality of the Cooper pairing. Hence, the observation of the particle density depletion and the measurement of the angular momentum will provide the information on the core-bound state and $p$-wave superfluidity. Moreover, the details on the zero energy Majorana state are discussed in the vicinity of the BCS-to-BEC evolution. It is demonstrated numerically that the zero energy Majorana state appears in the weak coupling BCS limit only when the vortex winding number is odd. There exist the $\kappa$ branches of the core bound states for a vortex state with vorticity $\kappa$, whereas only one of them can be the zero energy. This zero energy state vanishes at the BCS-BEC topological phase transition, because of interference between the core-bound and edge-bound states.

Abstract:
We show that two new intra-species P-wave superfluid phases appear in two-component asymmetric Fermi systems with short-range S-wave interactions. In the BEC limit, phonons of the molecular BEC induce P-wave superfluidity in the excess fermions. In the BCS limit, density fluctuations induce P-wave superfluidity in both the majority and the minority species. These phases may be realized in experiments with spin-polarized Fermi gases.

Abstract:
In a trapped atomic Fermi gas, one can tune continuously via a Feshbach resonance the effective pairing interaction between fermionic atoms from very weak to very strong. As a consequence, the low temperature superfluidity evolves continuously from the BCS type in the weak interaction limit to that of Bose-Einstein condensation in the strong pairing limit, exhibiting a BCS-BEC crossover. In this paper, we review recent experimental progress in atomic Fermi gases which elucidates the nature of the superfluid phase as the interaction is continuously tuned. Of particular interest is the intermediate or crossover regime where the $s$-wave scattering length diverges. We will present an intuitive pairing fluctuation theory, and show that this theory is in quantitative agreement with existing experiments in cold atomic Fermi gases.

Abstract:
Superconductivity and superfluidity of fermions require, within the BCS theory, matching of the Fermi energies of the two interacting Fermion species. Difference in the number densities of the two species leads either to a normal state, to phase separation, or - potentially - to exotic forms of superfluidity such as FFLO-state, Sarma state or breached pair state. We consider ultracold Fermi gases with polarization, i.e. spin-density imbalance. We show that, due to the gases being trapped and isolated from the environment in terms of particle exchange, exotic forms of superfluidity appear as a shell around the BCS-superfluid core of the gas and, for large density imbalance, in the core as well. We obtain these results by describing the effect of the trapping potential by using the Bogoliubov-de Gennes equations. For comparison to experiments, we calculate also the condensate fraction, and show that, in the center of the trap, a polarized superfluid leads to a small dip in the central density difference. We compare the results to those given by local density approximation and find qualitatively different behavior.

Abstract:
Strong evidence for pairing and superfluidity has recently been found in atomic Fermi gases at the BCS-BEC crossover both in collective modes and RF excitation energies. It is argued that the scale for the effective pairing gaps measured in RF experiments is set by the lowest quasiparticle in-gap excitation energies. These are calculated at the BCS-BEC crossover from semiclassical solutions to the Bogoliubov-deGennes equations. The strong damping of the radial breathing mode observed in the BCS limit occur when the lowest quasiparticle excitation energies coincide with the radial frequency, which indicates that a coupling between them take place.

Abstract:
A review of recent BEC-BCS crossover experiments in ultracold Fermi gases is given with particular emphasis on the work performed with lithium-6 at the University of Innsbruck.

Abstract:
We study the superfluid state of atomic Fermi gases using a BCS-BEC crossover theory. Our approach emphasizes non-condensed fermion pairs which strongly hybridize with their (Feshbach-induced) molecular boson counterparts. These pairs lead to pseudogap effects above $T_c$ and non-BCS characteristics below. We discuss how these effects influence the experimental signatures of superfluidity.