Induced interaction and crystallization of self-localized impurity fields in a Bose-Einstein condensate

We model the behavior of N classical impurity fields immersed in a larger Bose-Einstein condensate by N+1 coupled nonlinear Schroedinger 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.