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arXiv:2509.18423 · Physical Review X

Observation of Synchronization between Two Quantum van der Pol Oscillators in Trapped Ions

Jiarui Liu, Qiming Wu, Joel E. Moore, Hartmut Häffner, Christopher W. Wächtler

Published Apr 27, 2026• 3 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/w1bm-wjl4

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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.
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.
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.
Limit-cycle oscillators are the basic building blocks for synchronization; yet, the notion of a quantum limit cycle has remained unclear. Here, we study quantum limit cycles and synchronization in the presence of continuous heterodyne measurement. The resulting quantum trajectories, i.e., time evolutions of the quantum state conditioned on the measurement outcome, make the quantum limit cycles apparent. We focus on the paradigmatic model of the quantum van der Pol oscillator and on two-level systems. Our work provides insights into limit cycles in quantum systems, emphasizing their similarity to classical limit cycles subject to noise. Additionally, we connect theoretical measures of quantum synchronization to quantities experimentally accessible via heterodyne detection.
Van der Pol oscillators are prototypical self-sustaining oscillators which have been used to model nonlinear processes in biological and other classical processes. In this work, we investigate how quantum fluctuations affect phase-locking in one or many van der Pol oscillators. We find that phase-locking is much more robust in the quantum model than in the equivalent classical model. Trapped-ion experiments are ideally suited to simulate van der Pol oscillators in the quantum regime via sideband heating and cooling of motional modes. We provide realistic experimental parameters for 171Yb+{}^{171}\text{Yb}^+171Yb+ achievable with current technology.
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.
Recently, several studies have investigated synchronization in quantum-mechanical limit-cycle oscillators. However, the quantum nature of these systems remained partially hidden, since the dynamics of the oscillator's phase was overdamped and therefore incoherent. We show that there exist regimes of underdamped and even quantum-coherent phase motion, opening up new possibilities to study quantum synchronization dynamics. To this end, we investigate the Van der Pol oscillator (a paradigm for a self-oscillating system) synchronized to an external drive. We derive an effective quantum model which fully describes the regime of underdamped phase motion and additionally allows us to identify the quality of quantum coherence. Finally, we identify quantum limit cycles of the phase itself.