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arXiv:2608.29185

Second harmonic phase locking and synchronization blockade in quadratically coupled driven quantum van der Pol oscillators

Nissi Thomas, M. Senthilvelan

quant-phSubmitted Aug 29, 2026• Announced Sep 1, 2026
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Abstract

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.

Comments
accepted for publication in Physical Review A

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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 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 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.
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.
Scalable methods for detecting and quantifying the nonclassical nature of a quantum state in noisy environments are challenging due to a complex relationship between noise and quantum coherence. In particular, identifying experimentally accessible signatures of synchronization in such regimes remains an open problem. By leveraging promising experimental implementation, we underpin what possible direct measures of nonclassicality are available. This work outlines accessing quantum synchronization (QS) in the steady state of a driven quantum van der Pol oscillator (vdPo) using two distinct figures of merit: (i) the nonclassical area δδδ and (ii) the second-order correlation function g(2)(0)g^{(2)}(0)g(2)(0), both of which are viable in experimental architectures. The nonclassical area quantifier based on homodyne tomography allows us to assess the nonclassical nature of the vdPo state directly from the tomogram without requiring full state reconstruction or Wigner function negativity. Within a well-defined parameter regime of drive strength and detuning, both δδδ and g(2)(0)g^{(2)}(0)g(2)(0) exhibit pronounced signatures of synchronization that complements the phase coherence between the drive and the vdPo. We derive an analytical expression for the steady state density matrix and the corresponding tomogram of the system, valid for arbitrary strengths of the harmonic drive. Analysis of the quantum tomogram uncovers clear phase locking behaviour, enabling the identification of the synchronization region (Arnold tongue) directly in terms of g(2)(0)g^{(2)}(0)g(2)(0) and δδδ. Furthermore, the behaviour of g(2)(0)g^{(2)}(0)g(2)(0) provides a statistical perspective that reinforces the tomographic signatures of QS. By analyzing the interplay between the aforementioned metrics, our findings indicate a scalable and experimentally relevant framework for characterizing QS in the driven vdPo.
We study quantum van der Pol oscillator in which single-photon pumping competes with a nonlinear two-photon dissipative channel to produce a quantum limit cycle, and the question of how two such oscillators with non-identical intrinsic frequencies synchronize in the sense of phase locking when coupled dissipatively to one another. For a single oscillator, starting from a Lindblad description of the dynamics, we derive the exact Wigner-function evolution equation, and show that its Fokker--Planck approximation admits in the appropriate regime an equivalent classical Itô stochastic description that reproduces remarkably well the full Lindblad dynamics. Notably, the stochastic approach provides an efficient means of capturing system dynamics in the semiclassical regime, where a large number of energy eigenstates contribute and direct numerical simulations become computationally demanding. Restricting to the three lowest Fock states and using a parity symmetry, we obtain in closed form the stationary-state density matrix and its Wigner-function representation that displays pronounced nonclassical features. For two non-identical oscillators, we study the full Lindblad dynamics numerically and characterize synchronization through the stationary-state order parameter as a function of frequency mismatch and dissipative coupling. We find that increasing the effective nonlinear dissipative strength suppresses the synchronized region. Restricting the two oscillators to the three-level qutrit subspace, a parity symmetry reduces the stationary-state problem to an exact 414141-dimensional linear system, allowing to compute the synchronization order parameter. In this reduced description, results in the limit of large dissipation agree closely with those from the full dynamics, while exhibiting in the limit of small dissipation a double-peak synchronization structure accompanied by a phase reversal.
Classical self-sustained oscillators, which generate periodic motion without periodic external forcing, are ubiquitous in science and technology. The realization of nonclassical self-oscillators is an important goal of quantum physics. We here present the experimental implementation of a quantum van der Pol oscillator, a paradigmatic autonomous quantum driven-dissipative system with nonlinear damping, using a single trapped atom. We demonstrate the existence of a quantum limit cycle in phase space in the absence of a drive and the occurrence of quantum synchronization when the nonlinear oscillator is externally driven. We additionally show that synchronization can be enhanced with the help of squeezing perpendicular to the direction of the drive and, counterintuitively, linear dissipation. We also observe the bifurcation to a bistable phase-space distribution for large squeezing. Our results pave the way for the exploration of self-sustained quantum oscillators and their application to quantum technology.