  Driven classical self-sustained oscillators have been studied extensively in
the context of synchronization. Using the master equation, this work considers
the classically driven generalized quantum Rayleigh-van der Pol oscillator,
which is characterized by linear dissipative gain and loss terms as well as
three non-linear dissipative terms. Since two of the non-linear terms break the
rotational phase space symmetry, the Wigner distribution of the quantum
mechanical limit cycle state of the undriven system is, in general, not
rotationally symmetric. The impact of the symmetry-breaking dissipators on the
long-time dynamics of the driven system are analyzed as functions of the drive
strength and detuning, covering the deep quantum to near-classical regimes.
Phase localization and frequency entrainment, which are required for
synchronization, are discussed in detail. We identify a large parameter space
where the oscillators exhibit appreciable phase localization but only weak or
no entrainment, indicating the absence of synchronization. Several observables
are found to exhibit the analog of the celebrated classical Arnold tongue; in
some cases, the Arnold tongue is found to be asymmetric with respect to
vanishing detuning between the external drive and the natural oscillator
frequency.
