  We investigate quantum synchronization theoretically in a system consisting
of two cold ions in microtraps. The ions' motion is damped by a standing-wave
laser whilst also being driven by a blue-detuned laser which results in
self-oscillation. Working in a non-classical regime, where these oscillations
contain only a few phonons and have a sub-Poissonian number variance, we
explore how synchronization occurs when the two ions are weakly coupled using a
probability distribution for the relative phase. We show that strong
correlations arise between the spin and vibrational degrees of freedom within
each ion and find that when two ions synchronize their spin degrees of freedom
in turn become correlated. This allows one to indirectly infer the presence of
synchronization by measuring the ions' internal state.
