  The entanglement properties of quantum synchronization, based on a single-ion
phonon laser subjected to an external drive, have been studied. It is found
that the maximum value of steady-state entanglement between the ion's internal
and external states occurs near the noiseless boundary from synchronization to
unsynchronization, accompanied by noticeable oscillatory behaviors during the
corresponding time evolution of entanglement. In addition, the later time
dynamics of entanglement also indicates the occurrence of frequency
entrainment, as evidenced by the strong consistency between the bending of the
observed frequency and the emergence of Liouvillian exceptional points (LEPs)
in the first two eigenvalues of the Liouvillian eigenspectrum. Moreover, the
emergence of LEPs, which is intimately associated with frequency entrainment,
should be widely observed in quantum synchronization and can be explored in
LEPs-based applications.
