We present our experimental setup to produce ultracold strongly correlated fermionic superfluids made of a two-component spin-mixture of 6Li atoms. Employing standard cooling techniques, we achieve quantum degeneracy in a single-beam optical dipole trap. Our setup is capable of generating spin-balanced samples at temperatures as low as T/TF = 0.1 containing up to 5 × 10^4 atomic pairs. We can access different superfluid regimes by tuning the interparticle interactions close to a broad magnetic Feshbach resonance. In particular, we are able to explore the crossover from the molecular Bose-Einstein condensate (BEC) to the Bardeen-Cooper-Schrieffer (BCS) superfluid regimes. In the near future, we plan to study different collective excitations in these superfluid samples.

Using the laws of thermodynamics together with empirical data, we present a qualitative geometric construction of the fundamental relation of a pure substance S = S(E, N, V ), with S entropy, E energy, N number of particles and V volume. We analyze two very general type of substances, a "normal" and a "quantum" one, the main difference between them being that the latter presents superfluid phases. It is found that the constant entropy level curves are completely different in both cases, in the normal substances being obtuse while acute in quantum ones. A concomitant signature of the previous result is that the chemical potential can be both positive and negative in quantum substances, but only negative in normal ones. Our results suggest the existence of a region in the quantum substances that may be identified as a supersolid phase. We also make emphasis on the relevance of the present study within the context of superfluidity in ultracold gases.

The dynamics of Bose-Einstein condensation in a three-dimensional harmonic trap is studied explicitly including the Bogoliubov approximation for temperatures below the critical one. To model the evolution towards equilibrium at each cooling step, we derive quantum kinetic equations that describe the dynamics of the gas for temperatures above and below the transition temperature. These equations, valid in the Born and Markov approximations, consider the essential role of the chemical potential as the main parameter that signals the transition. The kinetic equation that describes the growth of the condensate below the transition temperature is derived within the Bogoliubov approximation. To illustrate our results we propose an energetic cooling protocol and simulate the whole sequence of the formation of a condensate.