Supersolid spectroscopy

We develop a linear-response theory to provide a unified description of two recent spectroscopy protocols for probing one-dimensional supersolid states realized in cold-atom systems. Both protocols involve applying a periodic optical potential to excite the supersolid and determine its excitation frequencies and density response characteristics. This information can be used to estimate the superfluid fraction. We validate our linear-response theory against nonlinear mean-field simulations of the dynamics for ring, box, and harmonically trapped cases. A key focus is the behavior at the band edge---the regime occurring when the optical potential used to excite the system has a wavelength that is twice the value of the supersolid lattice constant. Here symmetry can be used to selectively excite a mode from one of the two low-energy gapless excitation bands. We also consider the application of the spectroscopy protocols to determine the superfluid fraction, showing the relationship to hydrodynamic theory and a Josephson-junction array model. Our analysis and results suggest that the phase protocol we introduce has potential advantages for experimental application over the density protocol.