  Ultranarrow-linewidth atoms coupled to a lossy optical cavity mode
synchronize, i.e. develop correlations, and exhibit steady-state superradiance
when continuously repumped. This type of system displays rich collective
physics and promises metrological applications. These features inspire us to
investigate if analogous spin synchronization is possible in a different
platform that is one of the most robust and controllable experimental testbeds
currently available: ion-trap systems. We design a system with a primary and
secondary species of ions that share a common set of normal modes of vibration.
In analogy to the lossy optical mode, we propose to use a lossy normal mode,
obtained by sympathetic cooling with the secondary species of ions, to mediate
spin synchronization in the primary species of ions. Our numerical study shows
that spin-spin correlations develop, leading to a macroscopic collective spin
in steady-state. We propose an experimental method based on Ramsey
interferometry to detect signatures of this collective spin; we predict that
correlations prolong the visibility of Ramsey fringes, and that population
statistics at the end of the Ramsey sequence can be used to directly infer
spin-spin correlations.
