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  • Open Access

Quantum limit cycles with continuous symmetries from coherent parametric driving: Exact solutions and many-body extensions

Sihan Chen1,* and Aashish A. Clerk2,†

  • *Contact author: sihanc@uchicago.edu
  • †Contact author: aaclerk@uchicago.edu
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Phys. Rev. Research 8, 033264 – Published 3 September, 2026

DOI: https://doi.org/10.1103/xb4r-9wxk

Abstract

There is widespread interest in many-body quantum systems that exhibit limit-cycle or time-crystalline behavior. An ideal quantum limit cycle would be realized using fully coherent driving (to minimize noise) and also have a continuous internal symmetry (to ensure generation of monochromatic radiation). While these two requirements may seem incompatible, we introduce in this work a large class of multimode bosonic limit-cycle models based on coherent parametric driving, which possess an O(N) continuous symmetry. Surprisingly, the full quantum dissipative steady state of these models can be found exactly. They exhibit rich physics, including steady-state entanglement, reduced phase diffusion, and the possibility of realizing quantum limit tori. The basic mechanism we identify provides a unified way to understand how coherent parametric driving can yield symmetry-enriched limit cycles, and also helps us understand related models where the relevant symmetries are weakly broken. The models we study are compatible with a range of different experimental platforms, including quantum optical setups and superconducting quantum circuits.

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Phys. Rev. Research 8, 033264– Published 3 September, 2026
  • Received 4 May 2026
  • Accepted 10 August 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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