A supersolid combines the seemingly antithetical properties of crystalline order and superfluidity, where the same particles that self-assemble into a rigid crystal can flow through the system without dissipation. More than fifty years after its conjecture by Gross and later by Andreev, Lifshitz and Chester (ALC) this enigmatic phase of matter continues to elude experimental verification. In recent years, promising observations in pressurized Helium have intensified experimental and theoretical activity. Yet, a physical system that unambiguously displays supersolidity has not been found thus far, and the mere existence of free-space supersolidity in any type of quantum solid remains an open question.
In this talk we will explore how this question can be addressed in ultracold gases. In particular, I will show that Bosons with long-range soft-core interactions support both Gross-type supersolidity as well as ALC-type supersolids due to a condensation of zero-point defects emerging in the ground state of a self-assembled crystal. A discussion of the underlying phase diagram sheds light on the basic mechanisms behind the emergence of the different types of supersolids, and reveals an unexpected role of quantum exchange. Generally, the possibility to control soft-core interactions in cold gases promises a versatile approach for exploring an interesting spectrum of many-body physics, including strongly correlated superfluids, various cluster crystals and quantum glasses. Prospects for realizing this intriguing setting with Rydberg atoms, polar molecules or interacting photons will also be discussed.