Ultracold atoms are excellent systems to quantum-simulate the physics of ideal condensed-matter systems. A fundamental tool for this investigation is represented by optical lattices, i.e. periodic potentials generated by laser standing waves. Quantum phase transitions can be induced and investigated thanks to the possibility of accurately tuning the interactions between the particles, their mobility in the lattice, the amount of disorder and the system dimensionality. We will review some of the recent developments achieved at LENS in this field by studying ultracold quantum gases and mixtures in optical lattices.

We will focus on the possibility of introducing short-scale inhomogeneities in the lattice, which allows to study the interplay of interactions and disorder in the superfluid-insulator transition, one of the central problems of contemporary condensed-matter theory. We will discuss the observation of Anderson localization for a non-interacting Bose-Einstein condensate in a disordered optical lattice and the ongoing research to study the physics of localization in the presence of interactions between the atoms.

In order to characterize the properties of the different quantum phases novel diagnostic techniques have to be implemented. Recently, inelastic light scattering (Bragg spectroscopy) has allowed to study the excitations of 1D bosonic gases across the superfluid to Mott insulator transition. We will discuss this technique and the perspectives of using this tool to study the physics of disordered and strongly correlated 1D systems.