  We study the synchronization of a van der Pol self-oscillator with Kerr
anharmonicity to an external drive. We demonstrate that the anharmonic,
discrete energy spectrum of the quantum oscillator leads to multiple resonances
in both phase locking and frequency entrainment not present in the
corresponding classical system. Strong driving close to these resonances leads
to nonclassical steady-state Wigner distributions. Experimental realizations of
these genuine quantum signatures can be implemented with current technology.
