  We implement nonlinear anharmonic interaction in the coupled van der Pol
oscillators to investigate the quantum synchronization behaviour of the
systems. We study the quantum synchronization in two oscillator models, coupled
quantum van der Pol oscillators and anharmonic self-oscillators. We demonstrate
that the considered systems exhibit a high-order synchronization through
coupling in both classical and quantum domains. We show that due to the
anharmonicity of the nonlinear interaction between the oscillators the system
exhibits phonon blockade in the phase locking regime which is a pure
nonclassical effect and has not been observed in the classical domain. We also
demonstrate that for coupled anharmonic oscillators the system shows a multiple
resonance phase locking behaviour due to nonlinear interaction. We point out
that the synchronization blockade arises due to strong anticorrelation between
the oscillators which leads to phonon antibunching in the same parametric
regime. In the anharmonic oscillator case we illustrate the simultaneous
occurrence of bunching and antibunching effects as a consequence of
simultaneous negative and positive correlation between the anharmonic
oscillators. We examine the aforementioned characteristic features in the
frequency entrainment of the oscillators using power spectrum where one can
observe normal mode splitting and Mollow triplet in the strong coupling regime.
Finally, we propose a possible experimental realization for the considered
system in trapped ion and optomechanical setting.
