Order parameter and detection for a finite ensemble of crystallized one-dimensional dipolar bosons in optical lattices

We explore the ground-state properties of a few bosons with dipole-dipole interactions in a one-dimensional optical lattice. For comparatively strong interactions, a transition from a Mott-insulator state to a crystal state occurs. Herein, we provide a detailed characterization and a detection protocol for the resulting crystal state. Using the eigenvalues of the reduced one-body density matrix, we define an order parameter that characterizes all the emergent states occurring in the finite-sized ensemble and agrees with the signatures obtained from the analysis of the one- and two-body densities. We further demonstrate that these few-body emergent states can be detected experimentally using the variance of single-shot measurements.