Abstract:
We present a concise review of the physics of ultra-cold dipolar gases, based mainly on the theoretical developments in our own group. First, we discuss shortly weakly interacting ultra-cold trapped dipolar gases. Dipolar Bose-Einstein condensates exhibit non-standard instabilities and the physics of both Bose and Fermi dipolar gases depends on the trap geometry. We focus then the second part of the paper on strongly correlated dipolar gases and discuss ultra-cold dipolar gases in optical lattices. Such gases exhibit a spectacular richness of quantum phases and metastable states, which may perhaps be used as quantum memories. We comment shortly on the possibility of superchemistry aiming at the creation of dipolar heteronuclear molecules in lattices. Finally, we turn to ultra-cold dipolar gases in artificial magnetic fields, and consider rotating dipolar gases, that provide in our opinion the best option towards the realization of the fractional quantum Hall effect and quantum Wigner crystals.

Abstract:
We study the equilibrium current density profiles of harmonically trapped ultra-cold Fermi gases in quantum Hall-like states that appear when the quasi-two-dimensional trap is set in fast rotation. The density profile of the gas (in the rotating reference frame) consists of incompressible strips of constant quantized density separated by compressible regions in which the density varies. Remarkably, we find that the atomic currents flow in opposite directions in the compressible and incompressible regions -- a prediction that should be amenable to experimental verification.

Abstract:
Recent experiments with ultra-cold atoms have demonstrated the possibility of realizing experimentally fermionic superfluids with imbalanced spin populations. We discuss how these developments have shed a new light on a half- century old open problem in condensed matter physics, and raised new interrogations of their own.

Abstract:
Ultra-cold atomic systems provide a new setting where to investigate the role of long-range interactions. In this paper we will review the basics features of those physical systems, in particular focusing on the case of Chromium atoms. On the experimental side, we report on the observation of dipolar effects in the expansion dynamics of a Chromium Bose-Einstein condensate. By using a Feshbach resonance, the scattering length characterising the contact interaction can be strongly reduced, thus increasing the relative effect of the dipole-dipole interaction. Such experiments make Chromium atoms the strongest candidates at present for the achievement of the strong dipolar regime. On the theoretical side, we investigate the behaviour of ultra-cold dipolar systems in the presence of a periodic potential. We discuss how to realise this situation experimentally and we characterise the system in terms of its quantum phases and metastable states, discussing in detail the differences with respect to the case of zero-range interactions.

Abstract:
We briefly review the theory of Bose-Einstein condensation in the two-dimensional trapped Bose gas and, in particular the relationship to the theory of the homogeneous two-dimensional gas and the Berezinskii-Kosterlitz-Thouless phase. We obtain a phase diagram for the trapped two-dimensional gas, finding a critical temperature above which the free energy of a state with a pair of vortices of opposite circulation is lower than that for a vortex-free Bose-Einstein condensed ground state. We identify three distinct phases which are, in order of increasing temperature, a phase coherent Bose-Einstein condensate, a vortex pair plasma with fluctuating condensate phase and a thermal Bose gas. The thermal activation of vortex-antivortex pair formation is confirmed using finite-temperature classical field simulations.

Abstract:
Recent experiments on atom loss in ultra-cold Fermi gases all show a maximum at a magnetic field below Feshbach resonance, where the s-wave scattering length is large (close to inter-particle distance) and positive. These experiments have been performed over a wide range of conditions, with temperatures and trap depths spanning over three decades. Different groups have come up with different explanations, among them the emergence of Stoner ferromagnetism. Here, we show that this maximum is a consequence of two major steps. The first is the establishment of a population of shallow dimers, which is the combined effect of dimer formation through three-body recombination, and the dissociation of shallow dimers back to atoms through collisions. The dissociation process will be temperature dependent, and is affected by Pauli blocking at low temperatures. The second is the relaxation of shallow dimers into tightly bound dimers through atom-dimer and dimer-dimer collisions. We have constructed a simple set of rate equations describing these processes. Remarkably, even with only a few parameters, these equations reproduce the loss rate observed in all recent experiments, despite their widely different experimental conditions. Our studies show that the location of the maximum loss rate depends crucially on experimental parameters such as trap depth and temperature. These extrinsic characters show that this maximum is not a reliable probe of the nature of the underlying quantum states. The physics of our equations also explains some general trends found in current experiments.

Abstract:
The superfluid phases in the resonant dipolar Fermi gases are investigated by the standard mean-field theory. In contrast to the crossover from Bose-Einstein condensation (BEC) to Bardeen-Cooper-Schrieffer (BCS) superfluid in the Fermi gases with the isotropic interactions, the resonant dipolar interaction leads to two completely different BEC phases of the tight-binding Fermi molecules on both sides of the resonance, which are characterized by two order parameters with the distinct internal symmetries. We point that near the resonance, the two competitive phases can coexist, and an emergent relative phase between the two order parameters spontaneously breaks the time-reversal symmetry, which could be observed in the momentum resolved rf-spectroscopy.

Abstract:
We show how Fermi liquid theory can be applied to ultra-cold Fermi gases, thereby expanding their "simulation" capabilities to a class of problems of interest to multiple physics sub-disciplines. We introduce procedures for measuring and calculating position dependent Landau parameters. This lays the ground work for addressing important controversial issues: (i) the suggestion that thermodynamically, the normal state of a unitary gas is indistinguishable from a Fermi liquid (ii) that a fermionic system with strong repulsive contact interactions is associated with either ferromagnetism or localization; this relates as well to $^3$He and its p-wave superfluidity.

Abstract:
We investigate the Zel'dovich effect in the context of ultra-cold, harmonically trapped quantum gases. We suggest that currently available experimental techniques in cold-atoms research offer an exciting opportunity for a direct observation of the Zel'dovich effect without the difficulties imposed by conventional condensed matter and nuclear physics studies. We also demonstrate an interesting scaling symmetry in the level rearragements which has heretofore gone unnoticed.

Abstract:
After deriving the isothermal Hellmann-Feynman theorem (IHFT) that is suitable for mixed states in a canonical ensemble, we use this theorem to obtain the isothermal magnetic-field sweep theorems for the free, average and trapping energies, and for the entropy, specific heat, pressure and atomic compressibility of strongly-correlated ultra-cold quantum gases. In particular, we apply the sweep theorems to two-component Fermi gases in the weakly-interacting BCS and BEC limits, showing that the temperature dependence of the contact parameter can be determined by the variation of either the entropy or specific heat with respect to the scattering length. We also use the IHFT to obtain the Virial theorem in a canonical ensemble, and discuss its implications for quantum gases.