We report a normal-mode induced synchronization blockade in coupled quantum van der Pol oscillators under the influence of external drive. In this mechanism, the coupling hybridizes the oscillator modes into spectrally split normal modes. The destructive interference between the transitions to these modes blocks synchronization. We find that this blockade can be controlled simply by tuning the coupling strength and detuning allowing dynamic manipulation of quantum synchronization through collective mode dynamics. We analyze the phase-locking behaviour using perturbation analysis. Further, by deriving steady-state probability amplitudes we show how the energy redistribution and spectral splitting forms the basis of the blockade. Our results might provide new insights into how synchronization can be controlled in quantum systems.
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6 pages, 5 figures, accepted for publication in Physical Review A as a letter
We report a normal-mode induced synchronization blockade in coupled quantum van der Pol oscillators under the influence of external drive. In this mechanism, the coupling hybridizes the oscillator modes into spectrally split normal modes. The destructive interference between the transitions to these modes blocks synchronization. We find that this blockade can be controlled simply by tuning the coupling strength and detuning allowing dynamic manipulation of quantum synchronization through collective mode dynamics. We analyze the phase-locking behaviour using perturbation analysis. Further, by deriving steady-state probability amplitudes we show how the energy redistribution and spectral splitting forms the basis of the blockade. Our results might provide new insights into how synchronization can be controlled in quantum systems.
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. 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. A classical mean-field analysis of the corresponding equations of motion reproduces this asymmetric phase-locking geometry which 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. Furthermore, we show that the system exhibits mutual synchronization when both the oscillators satisfies 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.
We implement nonlinear anharmonic interaction in the coupled van der Pol oscillators to investigate the quantum synchronization behavior 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 behavior 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 a power spectrum where one can observe normal mode splitting and the Mollow triplet in the strong coupling regime. Finally, we propose a possible experimental realization for the considered system in the trapped ion and optomechanical setting.
Synchronization manifests itself in oscillators adjusting their frequencies and phases with respect to an external signal or another oscillator. In the quantum case, new features appear such as destructive interferences that can result in the suppression of phase locking. A three-level (spin-1) oscillator with equal gain and damping rates and subject to an external drive does not exhibit any 1:1 phase locking but 2:1 phase locking, i.e., its phase distribution features two maxima. This bistable locking at two opposite phases is a signature of the quantum interference synchronization blockade. An analogous behavior was found for two identical coupled spin-1 oscillators. In this work, we consider two coupled spin-1 oscillators and a drive applied to the first spin. This leads to two interference blockades between the drive and the first spin as well as between both spins. Although both interference blockades persist for strong drive and coupling strengths, remarkably, the undriven spin does show a 1:1 phase locking to the external drive. The magnitude of the locking is proportional to the drive strength if the drive strength is small. In other words, the undriven oscillator synchronizes to the external drive through both interference blockades while the blockades persist. For a chain of three coupled spin-1 oscillators, we find synchronization between the first and third spins mediated via the blockaded, second spin.
We investigate quantum synchronization in coupled self-sustained oscillators interacting via bosonic exchange. Using a phase-distribution-based synchronization measure, we identify bistable phase locking between 0 and $\ensuremath{\pi}$ and uncover an expanded Arnold-tongue structure under finite detuning, indicative of enhanced quantum phase locking. Power spectral analysis reveals dual-frequency locking associated with dressed-state formation, a uniquely quantum effect. This spectral splitting persists even without detuning and increases with coupling strength, signaling coherent hybridization of oscillator modes. A comparison with a classical approximation highlights key differences, including the absence of spectral doublets in the classical regime. Finally, we show that increasing nonlinear damping induces a sharp transition in the preferred phase difference and leads to decoherence through spectral peak broadening, underscoring the critical role of dissipation in shaping quantum synchronization dynamics.
We investigate how Kerr nonlinearity modifies quantum synchronization in a squeezed quantum van der Pol oscillator. We show that the Kerr interaction produces an amplitude-dependent frequency shift that drives a saddle-node bifurcation, transforming the classical phase-space structure from bistable to monostable dynamics. In the quantum regime, this transition manifests as systematic frequency pulling and spectral broadening, while the steady-state Wigner function reveals a continuous correspondence between the quantum state and the semiclassical attractor despite finite quantum fluctuations. By constructing global synchronization phase diagrams in the squeezing--Kerr parameter space, we uncover a remarkably linear dependence of the critical squeezing strength required to maintain phase locking on the Kerr nonlinearity. We further demonstrate that the synchronization boundary does not coincide with the crossover between super- and sub-Poissonian photon statistics, showing that synchronization and photon-number statistics characterize distinct aspects of the quantum steady state. These results provide quantitative design principles for controlling quantum synchronization through Kerr nonlinearity, with potential relevance to trapped-ion, superconducting-circuit, and optomechanical platforms.
Synchronization is a hallmark of collective behavior that emerges when nonlinear systems interact, spanning scales from mechanical oscillators to planetary orbits. As a universal phenomenon, it underpins the study of complex systems and has far-reaching technological implications. While classical synchronization has a long and rich history, it has not been observed experimentally between multiple quantum limit-cycle oscillators despite a decade of theoretical investigations. We realize synchronization between two quantum van der Pol oscillators by engineering dissipation in a mixed-isotope trapped-ion quantum simulator. The synchronized state is encoded in a fixed relative phase between the oscillators that is inaccessible to individual measurements and revealed only through joint readout of both oscillators, in stark contrast to the system in the (deterministic) classical limit where synchronization can be observed via individual phase measurements. We further show that the relative phase can be precisely controlled and that the chain of two oscillators can synchronize to an external field, suggesting applications in sensing. Our results provide a promising pathway for studying more complex synchronized quantum dynamics beyond two oscillators, where a theoretical treatment becomes increasingly challenging, and it remains to be understood whether genuinely quantum features persist in such cases.