The role of integrability in speeding up collective spin polarization.*

The central spin problem plays an important role in a variety of condensed matter systems, including quantum dots and nitrogen vacancy centers in diamond. We study an anisotropic central spin model with two classes of special eigenstates: bright and dark. We find these states to be robust against perturbations around two integrable points of the model. By driving of the central magnetic field, the structure of these eigenstates can be harnessed to access highly polarized many-body configurations. Using approximate shortcuts to adiabaticity, we develop a highly efficient and experimentally viable polarizing scheme.