  It is desirable to observe synchronization of quantum systems in the quantum
regime, defined by low number of excitations and a highly non-classical steady
state of the self-sustained oscillator. Several existing proposals of observing
synchronization in the quantum regime suffer from the fact that the noise
statistics overwhelms synchronization in this regime. Here we resolve this
issue by driving a self-sustained oscillator with a squeezing Hamiltonian
instead of a harmonic drive and analyze this system in the classical and
quantum regime. We demonstrate that strong entrainment is possible for small
values of squeezing, and in this regime the states are non-classical.
Furthermore, we show that the quality of synchronization measured by the FWHM
of the power spectrum is enhanced with squeezing.
