We investigate the dynamics of a quadratically coupled system under the influence of an external drive applied to the second oscillator, where the coupling facilitates a high-order synchronization with phase locking emerging in the form of $2:1$ between the oscillators. Our analysis reveals a synchronization blockade in the first oscillator, characterized by the complete suppression of conventional $1:1$ phase locking with the drive. Instead the first oscillator exhibits a $2:1$ phase-locking behavior arising from the anharmonicity induced by the quadratic coupling. In contrast, we observe that the directly driven second oscillator synchronizes with the drive, showing $1:1$ phase locking but notably at second-harmonic frequency (${\omega }_2=2{\omega }_d$). A classical mean-field analysis of the corresponding equations of motion reproduces this asymmetric phase-locking geometry, including the two symmetry-related 2:1 phase-locked states of the first oscillator and the 1:1 phase-locked state of the second oscillator. This demonstrates that the phase-locking structure itself can be understood from the nonlinear classical dynamics. The quantum analysis, however, reveals the microscopic origin of the synchronization blockade: the quadratic interaction imposes a two-phonon selection rule that suppresses the conventional single-phonon synchronization channel of the first oscillator. Furthermore, we show that the system exhibits mutual synchronization when both the oscillators satisfy the resonance condition, enabling coherent energy exchange facilitated by nonlinear quadratic coupling. The mutual synchronization shows synchronized regimes and also subtle suppression of synchronized regimes near resonance occurring due to spectral splitting of the energy states. Using perturbation analysis of the master equation within the low-excitation subspace, we analyze steady-state phase distribution and synchronization measures, supported by population statistics and spectral responses. We also propose possible experimental realizations in trapped ions and optomechanical setups. These findings highlight the crucial role of quadratic coupling in enabling nonclassical synchronization phenomena, offering deeper insights for quantum control strategies and the development of quantum information platforms.

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• Coupled oscillators
• Quantum optics
• Synchronization