Abstract. This article reviews the recent theoretical and experimental advances in the study of ultracold gases made of bosonic particles interacting via the longrange, anisotropic dipole-dipole interaction, in addition to the short-range and isotropic contact interaction usually at work in ultracold gases. The specific properties emerging from the dipolar interaction are emphasized, from the meanfield regime valid for dilute Bose-Einstein condensates, to the strongly correlated regimes reached for dipolar bosons in optical lattices.

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We study the expansion of a dilute ultracold sample of fermions initially trapped in a anisotropic harmonic trap. The expansion of the cloud provides valuable information about the state of the system and the role of interactions. In particular the time evolution of the deformation of the expanding cloud behaves quite differently depending on whether the system is in the normal or in the superfluid phase. For the superfluid phase, we predict an inversion of the deformation of the sample, similarly to what happens with Bose-Einstein condensates. Viceversa, in the normal phase, the inversion of the aspect ratio is never achieved, if the mean field interaction is attractive and collisions are negligible.Since the first experiments on trapped Bose-Einstein condensates the imaging of the expanding cloud, following the sudden switching off of the confining potential, has provided crucial information on the novel features exhibited by atomic gases in conditions of quantum degeneracy. These include, in particular, the bimodal structure of an expanding Bose gas at finite temperature and the anisotropy of the asymptotic profile of the condensate [1,2]. In the Thomas-Fermi limit, where the Gross-Pitaevskii equations coincide with the hydrodynamic theory of superfluids, the anisotropy of the expanded gas reflects the anisotropy of the pressure force which is stronger in the direction of tighter confinement [3]. The predictions of the hydrodynamic equations and of the consequent scaling behaviour exhibited by BoseEinstein condensates during the expansion have been investigated by several authors ( [4], [5], [6]), providing excellent agreement with experiments [7,8] and pointing out the difference with respect to the expansion of a non condensed gas. In the latter case, in the collisionless regime, the density profile approaches an isotropic shape, independent of the initial deformation of the gas.In this letter we study the problem of the expansion of an ultracold sample of fermions initially trapped in an anisotropic harmonic trap. We will show that also in the case of fermions the expansion of the gas provides valuable information about the state of the system and the role of interactions. We will consider a gas of atoms interacting with attractive forces. This is a natural requirement for the realization of Cooper-pairs and hence for the achievement of the superfluid phase [9]. Such interactions are naturally present in some fermionc species like 6 Li and can otherwise be obtained by changing the scattering length profiting of the presence of a Feshbach resonance.The description of the expansion of a cold fermionic gas in the normal and superfluid phase requires different theoretical approaches. For the normal phase we use the formalism of the Landau-Vlasov equations, while in the superfluid phase we study the expansion using the hydrodynamic theory of superfluids.We consider the case of two different fermionic states, hereafter called 1 and 2, initially confined in a harmonic trap. We assume that the two species are pres...

Abstract. This tutorial is a theoretical work, in which we study the physics of ultra-cold dipolar bosonic gases in optical lattices. Such gases consist of bosonic atoms or molecules that interact via dipolar forces, and that are cooled below the quantum degeneracy temperature, typically in the nK range. When such a degenerate quantum gas is loaded into an optical lattice produced by standing waves of laser light, new kinds of physical phenomena occur. These systems realize then extended Hubbard-type models, and can be brought to a strongly correlated regime. The physical properties of such gases, dominated by the long-range, anisotropic dipole-dipole interactions, are discussed using the meanfield approximations, and exact Quantum Monte Carlo techniques (the Worm algorithm).