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Quantum limit cycles and synchronization from a measurement perspective

Tobias Nadolny and Christoph Bruder

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Phys. Rev. Research 8, 023050 – Published 15 April, 2026

DOI: https://doi.org/10.1103/c5wj-4s73

Abstract

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.

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Phys. Rev. Research 8, 023050– Published 15 April, 2026
  • Received 14 November 2025
  • Accepted 18 March 2026
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

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