When a dilute gas of neutral fermionic atoms, trapped in a magnetic or optical confining potential, is cooled down to temperatures of a few tens of nanokelvins above absolute zero, the gas undergoes a phase transition. The fermionic atoms form pairs, that can flow as in an ideal fluid, without friction. This phenomenon is briefly introduced in the first chapter.

Such superfluid" state also occurs for bosonic atoms, that however do not need to pair. For Bose systems, Gross and Pitaevskii developed a successful description of the superfluid state based on a macroscopic wavefunction. Unlike the many-particles wavefunction, which depends on the positions of all atoms in the system, the macroscopic wavefunction depends only on one position coordinate. Yet it reliably encodes many aspects of the behaviour of the superfluid. The modulus squared of the macroscopic wavefunction is interpreted as the density of superfluid particles, while the gradient of the phase is linked to the velocity field. The superfluid properties follow from the partial di_erential equation that this macroscopic wavefunction must satisfy. This di_erential equation is known for superfluid Bose gases, but up to now there was no counterpart for fermionic system. The goal of this thesis is to develop a description of superfluidity in fermionic systems in terms of a macroscopic wavefunction and to employ it to study related phenomena such as dark solitons in Fermi superfluids.

In this thesis an effective field theory (EFT) suitable to describe the superfluid phase of an ultracold system of neutral fermionic atoms in a wide range of interaction and temperature configurations is developed in the framework of the path-integral formulation of quantum field theory [1]. At the heart of the EFT lies the hypothesis that the order parameter varies slowly in both time and space. The calculations that, from this weak assumption, lead to the final form of the EFT action are carried out in full detail. The EFT is then applied to the study of various aspects of Fermi superfluids in the BEC-BCS crossover interaction regime. By introducing fluctuations beyond mean field the spectrum of the collective excitations and the corrections to the critical temperature are evaluated, and the results are compared to those of other theoretical approaches.

Motivated by the interest gathered in recent years by the BEC polaron problem, an analogous system where the Bose-Einstein condensate is replaced by a superfluid fermionic gas is treated, and the corrections to the polaronic coupling constant and effective mass due to the interaction of an impurity with the collective excitations of the superfluid are evaluated [2]. The interaction dependence of the dispersion relations for the collective modes enables to extend the analogy with the BEC polaron system, that in principle would be limited to the BEC limit, to a wider region of the BEC-BCS crossover.

The EFT is then applied to the study of various aspects of dark solitons in ultracold Fermi gases. At first the stable soliton solutions in (quasi-)1D are studied and the effects of interaction, temperature, and imbalance on the density profiles and dynamics are precisely characterised [3]. The main finding in this context is the fact that the soliton core is an energetically favorable place where the unpaired particles { present in the system because of a nonzero population imbalance and/or finite temperature { can be accommodated. Next the snake instability mechanism, responsible for the decay of dark solitons in 3D, is considered. The spectrum of the unstable modes is examined and compared to the results of other theoretical approaches [4]. The minimum size that the system can have in order for the soliton to be stable is estimated and the behaviour of this quantity across the BEC-BCS crossover is compared to other data found in literature. In the BEC regime the EFT gives results in very good agreement with those of both the time-dependent Bogoliubov-de Gennes (TDBdG) simulations and of the coarse grained Bogoliubov-de Gennes theory. Moreover, it appears that the EFT is the only treatment that correctly describes the change in the relevant length scale, from the healing length in the BEC regime, to the pair coherence length in the BCS regime. The effects of imbalance on the soliton stability are also examined, finding that for a fixed interaction strength, the critical size necessary to avoid decay through the snake instability is larger for an imbalanced system than for a balanced one. In principle this provides experimentalists with a method to stabilise solitons by increasing the imbalance without being forced to reduce the dimensionality of the cloud.

The description we develop in this thesis opens the way to many applications. Where other models, such as the Bogoliubov-de Gennes theory, become computationally demanding even for a single vortex or soliton, the current description has the advantage of allowing a rapid implementation. Thus, in the future it will be possible to investigate the behaviour of the system when it contains many vortices or solitons { similar as for superconductors we can characterise vortex matter and learn to manipulate vortices and solitons. Also, the theory can be easily extended to multi-component fermionic superfluids, which allows us to investigate whether new phenomena { that do not occur in the individual superfluids { can instead occur in such mixtures.