What can be learnt on a supersolid from its high-energy scattering response? Analytical considerations

High-energy scattering spectroscopy is a widely-established technique for probing the characteristic properties of complex physical systems. Motivated by the recent observation of long-sought supersolid states in dipolar quantum Bose gases, I investigate the general relationships existing between the density contrast, the superfluid fraction, and the response to a high-energy scattering probe of such exotic states within a simple model. I focus on the scaling laws relating these three observables in the two extreme regimes of "shallow" and "deep" supersolids, which are of particular interest in describing the phase transitions of the supersolid to a uniform superfluid and an incoherent crystal state, respectively. I illustrate and specify the general formulae using relevant ansatzes for the supersolid states of dipolar gases in these regimes. This work brings an intuitive understanding of a concomitant study based on experiments and mean-field numerical simulations. It generally highlights new possibilities for probing supersolid states while providing a simple framework to describe such intriguing states.