Realization of a beat-note optical lattice for interferometry with Bose-Einstein condensates

In this thesis I report the realization of a large spacing optical lattice exploiting the beat-note of two retro-reflected laser beams with slightly different wavelengths i.e 1064 nm and 1013 nm. The resulting potential, in addition to a very fast modulation, exhibits a slow periodicity that provides an optical potential for the atoms that behaves like an effective lattice with 10 micrometers of separation between lattice sites. I studied theoretically the band structure of the new lattice and I performed numerical simulations in order to obtain predictions of the energy gap as a function of the lattice depth in different configurations. Regarding my experimental contribution, I have complitely designed and assembled the setup for the stabilization of the intensity and the frequency of the laser sources. In particular, for the stabilization of the relative frequencies of the two lasers, I develope a technique that allows to lock the two frequencies to the same optical cavity. Thanks to the high control of the residual forces on the sample and the capability to cancel the interatomic interactions using magnetic Feshbach resonaces, we have managed to observe spatial Bloch oscillations of condensates, measuring the atom number in each well. The observed dynamics exhibits a coherence up to 1 s that correponds to a sensitivity of the order of 5 times10^-5 g with a spatial resolution of ten microns. We characterized experimentally the frequency and the amplitude of the oscillations for different value of the force and the results show a very good agreement with theoretical simulations. To our knowledge the sensitivity acquired in this work is the higher reported in litterature in the measurement of small forces (<< g) using a trapped atom interferometer. The main limitations to the coherence time and the precision of the measurements are related to the limited control we have in the way we tune the external force, i.e, optically. The simplicity of the Bloch oscillation interferometer and the creation of a large spacing periodic potential open interesting perspective in the measurements of small and local forces with trapped condensates and can be useful for the investigation of Casimir-Polder forces or deviations of the gravitational law at short distances. In the first half of my PhD I participated to another project: we studied the behaviour of an ultra-cold mixture of two different internal states of potassium. In particular regimes of interaction the cloud tends to collapse but, at high density, many-body effects occur and stabilize the mixture to a fixed volume. In free space it doesn't collapse or expand and it behaves like a liquid droplet. We have characterized the time evolution and the equilibrium properties of the droplets and we have peformed experimental studies on the collisional dynamics between two droplets. Good agreement with theoretical analysis confirm our control on the system and open interesting perspective for the studies of their excitation spectrum and superfluid behaviour.