Similarities between alkali-metal gases in optical lattices with noninteger occupation of the lattice sites and quantum crystals are explored. The analogy with the vacancy liquid (VL) provides an alternative explanation to the Mott transition for the recent experiment on the phase transition in the lattice. The VL can undergo Bose-Einstein condensation (BEC) with Tc within experimental reach. Direct and vacancy-assisted mechanisms of the band motion for hyperfine impurities are discussed. A large concentration of vacancies can result in the spatial decomposition of the system into pure hyperfine components. Below the vacancy condensation the impurity component resembles 3He in 3He−−He II mixtures.
We model the behavior of N classical impurity fields immersed in a larger Bose-Einstein condensate by N+1 coupled nonlinear Schr\"odinger equations in one, two, and three space dimensions. We discuss the stability of the uniform miscible system and show the importance of surface tension for self-localization of the impurity fields. We derive analytically the attractive tail of the impurity-impurity interaction due to mediation by the underlying condensate. Assuming all impurity fields interact with the same strength, we explore numerically the resulting phase diagram, which contains four phases: (I) all fields are miscible; (II) the impurity fields are miscible with each other but phase separate from the condensate as a single bubble; (III) the localized impurity fields stay miscible with the condensate, but not with each other; and (IV) the impurity fields phase separate from the condensate and each other, forming a crystalline structure within a bubble. Thus, we show that a crystal can be constructed solely from superfluid components. Finally, we argue that the crystalline phases maintain their superfluid behavior, i.e., they possess a nonclassical rotational inertia, which---combined with lattice order---is a characteristic of supersolidity.