Spin dynamical decoupling for generating macroscopic superpositions of a free-falling nanodiamond

Levitated nanodiamonds containing negatively charged nitrogen-vacancy centers (${\text{NV}}^{-}$) have been proposed as a platform to generate macroscopic spatial superpositions. Requirements for this include having a long ${\text{NV}}^{-}$ spin coherence time, which necessitates formulating a dynamical decoupling strategy in which the regular spin flips do not cancel the growth of the superposition through the Stern-Gerlach effect in an inhomogeneous magnetic field. Here, we propose a scheme to place a $250$-nm-diameter diamond in a superposition with spatial separation of over $250$ nm, while incorporating dynamical decoupling. We achieve this by letting a diamond fall for $2.4$ m through a magnetic structure, including $1.13$ m in an inhomogeneous region generated by magnetic teeth.