We investigate two-component attractive Fermi gases with imbalanced spin populations in trapped one-dimensional configurations. The ground state properties are determined with the local density approximation, starting from the exact Bethe-ansatz equations for the homogeneous case. We predict that the atoms are distributed according to a two-shell structure: a partially polarized phase in the center of the trap and either a fully paired or a fully polarized phase in the wings. The partially polarized core is expected to be a superfluid of the Fulde-Ferrell-Larkin-Ovchinnikov type. The size of the cloud as well as the critical spin polarization needed to suppress the fully paired shell are calculated as a function of the coupling strength.

We determine the steady-state phases of a driven-dissipative Bose-Hubbard model, describing, e.g., an array of coherently pumped nonlinear cavities with a finite photon lifetime. Within a meanfield master equation approach using exact quantum solutions for the one-site problem, we show that the system exhibits a tunneling-induced transition between monostable and bistable phases. We characterize the corresponding quantum correlations, highlighting the essential differences with respect to the equilibrium case. We also find collective excitations with a flat energy-momentum dispersion over the entire Brillouin zone that trigger modulational instabilities at specific wavevectors.PACS numbers: 42.50. Ar,03.75.Lm,42.50.Pq,71.36.+c In recent years, the interest in the physics of quantum fluids of light in systems with effective photon-photon interactions has triggered many exciting investigations [1]. Some of the most remarkable features of quantum fluids, such as superfluid propagation [2,3] or generation of topological excitations [4][5][6][7] have been observed in experiments with solid-state microcavities. With the dramatic experimental advances in solid-state cavity and circuit quantum electrodynamics (QED), a considerable interest is growing on the physics of controlled arrays of nonlinear cavity resonators, which can be now explored in state-of-art systems [8,9]. This opens the way to the implementation of non-equilibrium lattice models of interacting bosons, particularly when effective on-site photonphoton interactions are large enough to enter the strongly correlated regime [10][11][12][13]. In this kind of systems, it is possible to realize the celebrated Bose-Hubbard model [14] for photons or polaritons. Since the first theoretical proposal for implementing this model in optical systems [15][16][17], early works have been focused on phenomena close to the equilibrium Mott insulator-Superfluid quantum phase transition [18,19]. Strongly-non equilibrium effects have been addressed only more recently [20][21][22][23] particularly in the interesting driven-dissipative regime where the cavity resonators are excited by a coherent pump which competes with the cavity dissipation processes. In such non-equilibrium conditions, these open systems are driven into steady-state phases whose collective excitations can be extremely different from the equilibrium case. However, to the best of our knowledge, very little is known so far on these important properties for the non-equilibrium Bose-Hubbard model.In this Letter, we present comprehensive results for the steady-state phases and excitations of the drivendissipative Bose-Hubbard model in the case of homogeneous coherent pumping. The steady-state density matrix and expectation values of the relevant observables have been calculated with an efficient mean-field approach, based on exact analytical quantum optical solutions of the single-cavity problem. A rich diagram is shown with multiple steady-state phases, whose stability and complex energy excitation spectrum hav...