Pith. sign in

REVIEW 2 major objections 6 minor 107 references

This review establishes that quantum synchronization is a well-defined, experimentally realized phenomenon with a converging set of measures, and that its many-body form connects directly to continuous time crystals.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

T0 review · deepseek-v4-flash

2026-08-01 12:44 UTC pith:DXGOPFSP

load-bearing objection A comprehensive and useful review that deserves refereeing, but it has an unmarked verbatim block from a source paper and a steady-state equation that looks wrong, so it's not publishable as-is. the 2 major comments →

arxiv 2607.19328 v1 pith:DXGOPFSP submitted 2026-07-21 quant-ph cond-mat.quant-gascond-mat.stat-mechphysics.atom-phphysics.optics

Quantum Synchronization

Parvinder Solanki , Albert Cabot , Fernando Iemini , Federico Carollo , Midhun Krishna , Yeshma Ibrahim , Michal Hajdu\v{s}ek , Igor Lesanovsky
show 3 more authors
Rosario Fazio Roberta Zambrini Sai Vinjanampathy
This is my paper
classification quant-ph cond-mat.quant-gascond-mat.stat-mechphysics.atom-phphysics.optics PACS 05.45.Xt
keywords quantum synchronizationopen quantum systemsLindblad master equationLiouvillian spectrumcontinuous time crystalsvan der Pol oscillatorKuramoto modelphase locking
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The review is trying to establish that quantum synchronization is a coherent, experimentally realized phenomenon with a working toolbox of measures, and that it extends from few-body oscillators to many-body phases such as continuous time crystals. It argues that open quantum systems—systems driven by both Hamiltonian dynamics and dissipation—provide the natural setting, and that the same signatures (phase locking, frequency entrainment, spectral peaks) appear across trapped ions, atom-cavity devices, Rydberg gases, and superconducting circuits. A sympathetic reader is meant to come away convinced that quantum synchronization is a genuine resource for quantum technologies, from sensors to engines, and that its many-body form is deeply linked to time-translation-symmetry breaking.

Core claim

The central claim is that the adjustment of rhythms observed in classical oscillators survives in quantum systems, but with quantum-fluctuation broadening, and it can be quantified. The review catalogs measures—temporal correlations, squeezed-fluctuation bounds, quasi-probability localization, spectral linewidths, coherence distances—and applies them to systems with and without classical limit cycles, including finite-level systems where synchronization is defined through phase-space distributions. It also maps the many-body form: macroscopic synchronization (Kuramoto-like transitions) and continuous time crystals, where the collective oscillation in an open many-body system is the thermodyn

What carries the argument

The key object is the Lindblad master equation and its Liouvillian superoperator: the spectrum's imaginary eigenvalues encode oscillatory modes and its spectral gaps encode transient synchronization, while purely imaginary eigenvalues in the thermodynamic limit signal continuous time crystals. The review also leans on the Kuramoto order parameter for collective phase coherence and on Wigner/Husimi phase-space distributions for phase locking. These tools let the review move from a single driven oscillator to many-body phases on a common footing.

Load-bearing premise

The synthesis presupposes that the disparate phenomena it brings together—entrainment of oscillators, phase locking of finite-level systems, transient synchronization from spectral gaps, macroscopic ensemble coherence, and continuous time crystals—are all manifestations of one concept, 'quantum synchronization,' even though the review itself notes that no single correlation or measure is a distinctive signature of it.

What would settle it

If a pair of detuned quantum van der Pol oscillators is found in which the Pearson correlation of local observables is near unity (indicating synchronization) while the relative-phase Husimi-Q distribution remains uniformly flat (indicating no phase locking), then the two leading families of measures in the review would be contradicting each other; that would be a concrete experimental test of whether the reviewed 'synchronization' is one phenomenon or several.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • If the review's synthesis is right, quantum synchronization can be treated as a design resource: the same phase-locking signatures that appear in trapped-ion phonons can be engineered into sensors and thermal machines.
  • Continuous time crystals should be understood as the many-body, thermodynamic-limit limit of synchronization, so tools developed for one area (e.g., Liouvillian spectroscopy) transfer directly to the other.
  • The catalogued measures give experimentalists a menu for detecting synchronization without full state tomography, including real-time homodyne-current phase relations.
  • The classification of synchronization as stable versus metastable, finite versus full ensemble, and robust versus complete provides a common language for comparing results across different platforms.
  • The review's open problems—robust strictly-quantum time crystals and canonical measures—define a research agenda rather than a closed field.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • My inference, not the paper's claim: because the review itself notes that no single correlation or measure is a distinctive signature of synchronization, published claims of 'quantum synchronization' across different papers may not be quantitatively comparable; a measure derived directly from the Liouvillian spectrum could provide a common foundation.
  • A testable extension of the paper's connection between time crystals and synchronization is to use relative-phase Husimi marginals to detect continuous-time-crystal transitions in atom-cavity and Rydberg experiments, giving a synchronization-centric diagnostic of time-crystalline order.
  • The review's focus on Markovian dynamics suggests a stress test: rerun the reviewed few-body synchronization results under non-Markovian reservoirs, where the spectral-gap mechanism may fail and transient synchronization may become less robust.
  • A more speculative extension: the synchronization blockade effect, described as symmetry-enforced destructive interference of coherence channels, could be used as a tuneable switch in quantum networks, suppressing or allowing synchronization without changing the drive strength.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

2 major / 6 minor

Summary. This review surveys the growing literature on quantum synchronization. It opens with a classical-synchronization primer, introduces the open-quantum-systems toolkit, then catalogs synchronization measures (temporal, quantum, phase/frequency, Liouvillian-spectral, and emission-based). It reviews few-body systems, including quantum van der Pol oscillators, finite-level systems, synchronization blockade, transient and steady synchronization, and trajectory approaches; many-body systems, including quantum Kuramoto models, macroscopic synchronization, boundary time crystals and other continuous time-crystal models, correlations, and trajectories; and applications to thermodynamics and quantum technologies. The unifying theme is that synchronization persists in the quantum regime, but with quantum fluctuations, blockade phenomena, and new forms such as time-crystalline coherence.

Significance. As a review, this is a useful and generally reliable map of an active field. Its strengths are breadth (from classical background to recent experiments), a helpful classification of synchronization forms (Table I), and explicit discussion of caveats, notably the statement in §III.B that neither classical nor quantum correlations are in general distinctive signatures of synchronization. The authors are primary contributors, and the review is frank about open problems, such as the non-canonicity of measures. Because the paper's claim is descriptive rather than a new derivation, the skeptical concern about untested equivalence of measures does not, as worded, land: the review does not assert that Eqs. (17), (18), (22), and (27) are equivalent; it presents them as a toolbox. However, the review must be technically accurate in its equations, and two equations need correction.

major comments (2)
  1. [§V.F.1, Eq. (35)] The exact stationary state of the boundary time-crystal model is misprinted. With the master equation Eq. (31) containing the dissipation rate κ, the NESS must depend on ω_o/κ. Equation (35) as printed has no κ in η = Σ [S_-/(-iω_o N/2)]^j, making the steady state independent of κ and dimensionally inconsistent. The later trajectory discussion (§V.G, quoting Cabot et al.) uses β = ωN/(2κ), indicating that the intended denominator contains κ. Please correct the formula and re-verify its normalization and sign.
  2. [§IV.A.1, Eq. (25)] The mean-field reduction of Eq. (24) is written as ḍα = iω₀α + (γ₁/2)α − γ₂|α|²α. For α = ⟨â⟩, the coherent part of Eq. (24) gives ḍα = −iω₀α + ... ; the printed plus sign is inconsistent with the Hamiltonian term in Eq. (24). This error propagates into the claimed correspondence with the Stuart–Landau oscillator. Please correct the sign or state explicitly a convention in which α is not ⟨â⟩.
minor comments (6)
  1. [§III.B.5, Eq. (23)] The right-hand side of Eq. (23) integrates over 'dω' but contains e^{-iωτ}; the integration variable should be dτ. Please also define the time-limit variable consistently.
  2. [§IV.B, Eq. (27)] The notation for the SU(N)-based synchronization measure is confusing: N is used both for the Hilbert-space dimension and for the normalization constant, and the prefactor '2N/N' is likely '2/N'. Please clarify and check the formula against the cited work (Solanki et al., 2023).
  3. [§III.B.4, Eq. (28)] The spectral-gap inequality is misformatted; it should read |λ^{ˀRe}_{ˀk}| ≪ |λ^{Re}_k| for ˀk ≠ k, with the absolute values placed correctly.
  4. [§V.C, Table I] The last two columns/labels ('Robust', 'Complete') are difficult to parse in the typeset version. Please refine the labels and caption so the distinction is clear.
  5. [§III.B / §VIII] In light of the abstract's promise of 'measures that quantify them', I encourage a short paragraph explicitly stating that the different measures are not interchangeable and that no canonical measure has been established. This would directly address the natural question of whether the surveyed phenomena constitute a single concept.
  6. [General] Several inline citations appear malformed or lowercased, e.g., 'hai Li et al., 2026' in §V.C. Please check the bibliography production for such artifacts.

Circularity Check

0 steps flagged

No significant circularity: as a review, the paper catalogs and defines rather than derives; the few self-cited links are definitions or interpretive classifications, not fitted inputs or forced predictions.

full rationale

This is a review article, so there is no derivation chain in which a predicted quantity is constructed from the same data. The central synthesis—that disparate phenomena are forms of quantum synchronization—is supported by an explicit definition (Eq. 29: synchronized when local expectation values coincide with nonvanishing time-dependence) and by catalogued measures, experiments, and model analyses. No step reduces a prediction to a fit: the measures in Eqs. (17), (18), (22), and (27) are presented as alternative diagnostics, and the review explicitly concedes in Sec. III.B that 'neither classical nor quantum correlations are in general distinctive signatures of synchronization or dynamical locking.' This concession undermines any claim that the measures are canonical, but it is a caveat rather than a circular reduction. The closest candidate is Sec. V.D, where collective oscillations in boundary time crystals are said to 'constitute coherence synchronization in many-body open quantum systems (Solanki et al., 2022)' and are noted to fit Eq. (29). That is a definitional classification, not a derivation of a result from the definition; no externally predicted quantity is being smuggled in. Many citations are to the authors' own prior works, but those works are independent research results, and the review does not invoke them as a forbidden uniqueness theorem or ansatz. No self-citation is load-bearing in the sense of replacing an argument with an assertion that the present paper's own conclusion is true. The manuscript is self-contained as an organization of existing results, and no circular step meets the quoted-evidence threshold.

Axiom & Free-Parameter Ledger

0 free parameters · 5 axioms · 0 invented entities

This review introduces no free parameters and no invented entities; its explanatory burden is carried by the surveyed models' own parameters (γ1, γ2, F, ω_o, κ, g, V). The axioms listed are the background frameworks the review leans on to present the field as coherent: Markovian open-quantum-system dynamics, exactness of mean-field in the thermodynamic limit for collective models, transfer of classical nonlinear-dynamics results to quantum regimes, and standard spectral/regression theorems. None of these are new to this paper.

axioms (5)
  • domain assumption Markovian, weakly-coupled dynamics governed by GKSL master equations adequately describe the surveyed experimental platforms
    The entire survey (Secs. III–V) is framed in the Lindblad formalism (Eq. 12); Born–Markov approximations are invoked in §III.A.1 without re-derivation for each platform.
  • domain assumption Mean-field factorization becomes exact for all-to-all/collective models in the thermodynamic limit
    Invoked for boundary time crystals (§V.E.1: 'this approach is exact as N→∞') and relied on for the CTC phase diagram and for claims about the limits N→∞ and t→∞ not commuting.
  • standard math Watanabe–Oshikawa no-go theorem restricts closed-system time crystals, so open systems are the legitimate arena for continuous time crystals
    Used in §V.D to motivate boundary time crystals as an evasion of the no-go theorem; standard published result.
  • domain assumption Classical nonlinear-dynamics results (Kuramoto transition, Adler equation, Arnold tongues, Hopf bifurcations) transfer to the weakly dissipative quantum regime via semiclassical and phase-space mappings
    The quantum/classical correspondence in §IV.A relies on phase-space and mean-field limits; the review itself acknowledges deviations (smooth crossover replacing the saddle-node bifurcation in the deep quantum regime).
  • standard math Quantum regression theorem and Evans' theorem guaranteeing at least one Liouvillian steady state
    Used for power spectra (Eq. 20, §III.B.3) and for the existence of steady states (§III.A.1, 'guaranteed by Evans' theorem').

pith-pipeline@v1.3.0-alltime-deepseek · 50878 in / 21693 out tokens · 208736 ms · 2026-08-01T12:44:29.737886+00:00 · methodology

0 comments
read the original abstract

Natural and engineered classical systems are replete with examples of synchronization, understood as the adjustment of rhythms of physical systems. Such synchronization is at the heart of the stability of several classical technologies, such as mechanical bridges and electrical networks. Given the advent of quantum simulation and computation technologies, it is natural to study a quantum analogue of synchronization and explore novel applications. This review surveys synchronization in few and many-body quantum systems, measures that quantify them, and their applications to quantum technologies.

Figures

Figures reproduced from arXiv: 2607.19328 by Albert Cabot, Federico Carollo, Fernando Iemini, Igor Lesanovsky, Michal Hajdu\v{s}ek, Midhun Krishna, Parvinder Solanki, Roberta Zambrini, Rosario Fazio, Sai Vinjanampathy, Yeshma Ibrahim.

Figure 2
Figure 2. Figure 2: FIG. 2. Maximum value of the quantum Fisher information ρˆ= 1 ηˆηˆ † with ˆη = X S − ! (35) [PITH_FULL_IMAGE:figures/full_fig_p033_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: For short times, t & 1, the GMCs grow according lfbth hf thdl d kth d However, the genuine multipartite correlation cannot [PITH_FULL_IMAGE:figures/full_fig_p033_3.png] view at source ↗
Figure 1
Figure 1. Figure 1: FIG. 1. Signatures of the dynamical regimes in individual realizations. (a) Stationary value of the z component of the magnetization, mz FIG. 11 Quantum trajectories of continuous time crystals. (a) The plot shows the stationary phase diagram for the boundary [PITH_FULL_IMAGE:figures/full_fig_p035_1.png] view at source ↗

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

107 extracted references · 8 linked inside Pith

  1. [1]

    M., and S

    Abrams, D. M., and S. H. Strogatz (2004), Phys. Rev. Lett. 93, 174102. Acebr´ on, J. A., L. L. Bonilla, C. J. P´ erez Vicente, F. Ritort, and R. Spigler (2005), Rev. Mod. Phys.77,

    2004

  2. [4]

    V., and L

    Albert, V. V., and L. Jiang (2014), Phys. Rev. A89, 022118. Alicki, R., and J. Messer (1983), J. Stat. Phys.32(2),

    2014

  3. [16]

    Ne’eman, and A

    Bohm, A., Y. Ne’eman, and A. O. Barut (1988),Dynami- cal Groups and Spectrum Generating Algebras: Vol. 1, 2 (World Scientific, Singapore, Singapore). Booker, C., B. Buˇ ca, and D. Jaksch (2020), New J. Phys. 22(8), 085007. Braunstein, S. L., and C. M. Caves (1994), Phys. Rev. Lett. 72,

    1988

  4. [20]

    (2025), Sci

    Arumugam, D. (2025), Sci. Rep.15(1), 13446. Aspelmeyer, M., T. J. Kippenberg, and F. Marquardt (2014), Rev. Mod. Phys.86,

    2025

  5. [24]

    Chakraborty, T., and R. H. Rand (1988), Int. J. of Nonlinear Mech.23(5),

    1988

  6. [30]

    M¨ uller, K

    Czartowski, J., R. M¨ uller, K. ˙Zyczkowski, and D. Braun (2021), Phys. Rev. A104, 012410. Danner, L., C. Padurariu, J. Ankerhold, and B. Kubala (2021), Phys. Rev. B104, 054517. Das, B., R. Ghosh, and V. Mukherjee (2026), Phys. Rev. A 113(5), 052205. Deffner, S., and S. Campbell (2019),Quantum Thermody- namics: An introduction to the thermodynamics of qua...

    2021

  7. [38]

    Janson, D

    Balanov, A., N. Janson, D. Postnov, and O. Sosnovtseva (2009),From simple to complex(Springer). Bastidas, V. M. (2025), Phys. Rev. Res.7, 043029. Bastidas, V. M., I. Omelchenko, A. Zakharova, E. Sch¨ oll, and T. Brandes (2015), Phys. Rev. E92, 062924. Baur, D., S. Hertlein, A. Baumg¨ artner, J. Stefaniak, T. Esslinger, G. Natale, and T. Donner (2026), Phy...

    2009

  8. [50]

    (1992), Phys

    Kaneko, K. (1992), Phys. D55(3),

    1992

  9. [57]

    Galve, and R

    Cabot, A., F. Galve, and R. Zambrini (2017), New J. Phys. 19(11), 113007. Cabot, A., G. L. Giorgi, F. Galve, and R. Zambrini (2019), Phys. Rev. Lett.123, 023604. Cabot, A., G. L. Giorgi, and R. Zambrini (2021a), New J. Phys.23(10), 103017. Cabot, A., G. L. Giorgi, and R. Zambrini (2024b), PRX Quantum5, 030325. Cabot, A., G. Luca Giorgi, and R. Zambrini (2...

    Pith/arXiv arXiv 2017

  10. [58]

    Solanki, M

    Krishna, M., P. Solanki, M. Hajduˇ sek, and S. Vinjanampathy (2023), Phys. Rev. Lett.130(15), 150401. Krithika, V. R., P. Solanki, S. Vinjanampathy, and T. S. Mahesh (2022), Phys. Rev. A105, 062206. Kubo, R. (1962), J. Phys. Soc. Jpn.17(7),

    2023

  11. [62]

    Pappalardi (2020), Phys

    Lerose, A., and S. Pappalardi (2020), Phys. Rev. A102, 032404. Leyva, I., R. Sevilla-Escoboza, J. M. Buld´ u, I. Sendi˜ na Nadal, J. G´ omez-Garde˜ nes, A. Arenas, Y. Moreno, S. G´ omez, R. Jaimes-Re´ ategui, and S. Boccaletti (2012), Phys. Rev. Lett.108, 168702. Li, L., M. J. Hall, and H. M. Wiseman (2018), Phys. Rep. 759,

    2020

  12. [63]

    Es’haqi-Sani, X

    hai Li, S., N. Es’haqi-Sani, X. Li, and W. Li (2026), arXiv:2601.20186 [quant-ph]. Li, T., Z.-X. Gong, Z.-Q. Yin, H. T. Quan, X. Yin, P. Zhang, L.-M. Duan, and X. Zhang (2012), Phys. Rev. Lett.109, 163001. Li, W. (2022), Phys. Rev. A106, 023512. Li, W., C. Li, and H. Song (2017a), Phys. Rev. E95, 022204. Li, W., C. Li, and H. Song (2017b), Quantum Inf. Pr...

    arXiv 2026

  13. [70]

    Mukhopadhyay, R

    Montenegro, V., C. Mukhopadhyay, R. Yousefjani, S. Sarkar, U. Mishra, M. G. A. Paris, and A. Bayat (2025), Phys. Rep.1134,

    2025

  14. [71]

    Rivera, J

    Montoya, F., M. Rivera, J. Escalona, and P. Parmananda (2013), Phys. Lett. A377(43),

    2013

  15. [74]

    Vinjanampathy, and J

    Murtadho, T., S. Vinjanampathy, and J. Thingna (2023), Phys. Rev. Lett.131, 030401. Nadolny, T., and C. Bruder (2023), Phys. Rev. Lett. 131(19), 190402. Nadolny, T., and C. Bruder (2026), Phys. Rev. Res.8, 023050. Nadolny, T., C. Bruder, and M. Brunelli (2025), Phys. Rev. X15(1), 011010. Nagumo, J., S. Arimoto, and S. Yoshizawa (1962), Proc. IRE 50(10),

    2023

  16. [78]

    Pecora, L. M., F. Sorrentino, A. M. Hagerstrom, T. E. Mur- phy, and R. Roy (2014), Nat. Commun.5(1),

    2014

  17. [80]

    Piergentili, J

    Li, W., P. Piergentili, J. Li, S. Zippilli, R. Natali, N. Malossi, G. Di Giuseppe, and D. Vitali (2020), Phys. Rev. A101, 013802. Li, Y., C. Wang, Y. Tang, and Y.-C. Liu (2024), Phys. Rev. Lett.132(18), 183803. Li, Y., Z. Xie, X. Yang, Y. Li, X. Zhao, X. Cheng, X. Peng, J. Li, E. Lutz, Y. Lin, and J. Du (2025a), Sci. Adv.11(41), eady5649. Li, Y., X. Zhang...

    2020

  18. [91]

    Kongkhambut, H

    Skulte, J., P. Kongkhambut, H. Keßler, A. Hemmerich, L. Mathey, and J. G. Cosme (2024), Phys. Rev. A109(6), 063317. Solanki, P., A. Cabot, M. Brunelli, F. Carollo, C. Bruder, and I. Lesanovsky (2025), Phys. Rev. A112, L030601. Solanki, P., N. Jaseem, M. Hajduˇ sek, and S. Vinjanampathy (2022), Phys. Rev. A105, L020401. Solanki, P., M. Krishna, M. Hajduˇ s...

    2024

  19. [92]

    Sudler, A. J., J. Talukdar, and D. Blume (2024), Phys. Rev. E109, 054207. Sun, J. T., H. D. Liu, and X. X. Yi (2024), Phys. Rev. A 109, 023502. Tan, R., C. Bruder, and M. Koppenh¨ ofer (2022), Quantum 6,

    2024

  20. [93]

    Aronson, D. G., E. J. Doedel, and H. G. Othmer (1987), Phys. D25(1),

    1987

  21. [95]

    Velasco, V., and M. S. Neto (2021), J. Phys. Commun.5(1), 015003. Vinjanampathy, S., and J. Anders (2016), Contemp. Phys. 57(4),

    2021

  22. [97]

    Carollo, and I

    Cabot, A., F. Carollo, and I. Lesanovsky (2024a), Phys. Rev. Lett.132(5), 050801. 44 Cabot, A., F. Carollo, and I. Lesanovsky (2026), arXiv:2503.21753 [quant-ph]. Cabot, A., F. Galve, V. M. Egu ´ ıluz, K. Klemm, S. Maniscalco, and R. Zambrini (2018), npj Quantum Inf.4(1),

    arXiv 2026

  23. [99]

    Kosloff (2003), Phys

    Feldmann, T., and R. Kosloff (2003), Phys. Rev. E68, 016101. Ferioli, G., A. Glicenstein, I. Ferrier-Barbut, and A. Browaeys (2023), Nat. Phys.19(9),

    2003

  24. [100]

    Guarnieri, G., M. T. Mitchison, A. Purkayastha, D. Jaksch, B. Buˇ ca, and J. Goold (2022), Phys. Rev. A106, 022209. Gupta, S., A. Campa, and S. Ruffo (2014), J. Stat. Mech. 2014(8), R08001. Gupta, S., A. Campa, and S. Ruffo (2018),Statistical Physics of Synchronization(Springer, Cham, Switzerland). Ha, S.-Y., E. Jeong, and M.-J. Kang (2010), Nonlinearity 23(12),

    2022

  25. [101]

    (1926), London Edinburgh Dublin Philos

    van der Pol, B. (1926), London Edinburgh Dublin Philos. Mag. J. Sci.2(11),

    1926

  26. [106]

    Zheludev, N. I. (2024), Nat. Photon.18(11),

    2024

  27. [114]

    Pikovsky, and M

    Yeldesbay, A., A. Pikovsky, and M. Rosenblum (2014), Phys. Rev. Lett.112, 144103. Yin, X.-L., M. Guo, J. Huang, H.-w. J. Lee, and G. Zhang (2025), arXiv:2511.07134 [quant-ph]. Ypey, D., W. VanMeerwijk, C. Ince, and G. Groos (1980), J. Theor. Biol.86(4),

    arXiv 2014

  28. [119]

    Luoma, and W

    Link, V., K. Luoma, and W. T. Strunz (2019), Phys. Rev. A 99, 062120. Liu, B., L.-H. Zhang, Y. Ma, Q.-F. Wang, T.-Y. Han, J. Zhang, Z.-Y. Zhang, S.-Y. Shao, Q. Li, H.-C. Chen, G.- C. Guo, D.-S. Ding, and B.-S. Shi (2025a), Nat. Commun. 16(1),

    2019

  29. [131]

    Wang, H., A. L. Chudnovskiy, A. Gorsky, and A. Kamenev (2020), Phys. Rev. Res.2, 033025. Wang, J., S. Yang, Z. Wang, R. Qi, H. Hu, W. Li, and J. Jie (2026), arXiv:2603.14311 [quant-ph]. Wang, P., and R. Fazio (2021), Phys. Rev. A103(1), 013306. Wang, Z., R. Gao, X. Wu, B. Buˇ ca, K. Mølmer, L. You, and F. Yang (2025), Phys. Rev. Lett.135, 230401. Watanabe...

    arXiv 2020

  30. [137]

    (1946), Proc

    Adler, R. (1946), Proc. IRE34,

    1946

  31. [160]

    Ding, D., Z. Bai, Z. Liu, B. Shi, G. Guo, W. Li, and C. S. Adams (2024), Sci. Adv.10(9), eadl5893. Dogra, N., M. Landini, K. Kroeger, L. Hruby, T. Donner, and T. Esslinger (2019), Science366(6472),

    2024

  32. [167]

    Jiang, Y

    Jiao, Y., W. Jiang, Y. Zhang, J. Bai, Y. He, H. Shen, J. Zhao, and S. Jia (2025), Nat. Commun.16(1),

    2025

  33. [188]

    Nande, S. S., O. Lhamo, M. Paul, R. Bassoli, and F. H. Fitzek (2023a), in2023 IEEE aerospace conference(IEEE) pp. 1–9. Nande, S. S., M. Paul, S. Senk, M. Ulbricht, R. Bassoli, F. H. Fitzek, and H. Boche (2023b), Comput. Networks229, 109772. Nande, S. S., T. Rossi, M. I. Habibie, M. Barhoumi, K. Pala- parthy, W. Mansouri, A. Raju, R. Bassoli, E. Cianca, F....

    2024

  34. [200]

    Qian, J., G. Dong, L. Zhou, and W. Zhang (2012), Phys. Rev. A85(6), 065401. Raskatla, V., T. Liu, J. Li, K. MacDonald, and N. I. Zheludev (2024), Phys. Rev. Lett.133(13), 136202. Razzoli, L., M. Carrega, F. Cavaliere, G. Benenti, and M. Sassetti (2024), Quantum Sci. Technol.9(4), 045032. 48 Repp, B. H., and Y.-H. Su (2013), Psychon. Bull. Rev.20(3),

    2012

  35. [207]

    Esencan, M., A. I. Lvovsky, and B. Buˇ ca (2026), arXiv:2602.20269 [quant-ph]. Es’haqi-Sani, N., G. Manzano, R. Zambrini, and R. Fazio (2020), Phys. Rev. Res.2, 023101. Evans, D. E. (1977), Commun. Math. Phys.54,

    arXiv 2026

  36. [211]

    D’Amico, M

    Campbell, S., I. D’Amico, M. A. Ciampini, J. Anders, N. Ares, S. Artini, A. Auff` eves, L. Bassman Oftelie, L. P. Bettmann, M. V. S. Bonan¸ ca, T. Busch, M. Campisi, M. F. Cavalcante, L. A. Correa, E. Cuestas, C. B. Dag, S. Dago, S. Deffner, A. Del Campo, A. Deutschmann- Olek, S. Donadi, E. Doucet, C. Elouard, K. Ensslin, P. Erker, N. Fabbri, F. Fedele, G...

    2026

  37. [231]

    Bagheri, M., M. Poot, L. Fan, F. Marquardt, and H. X. Tang (2013), Phys. Rev. Lett.111, 213902. Bak, P. (1982), Rep. Prog. Phys.45(6),

    2013

  38. [262]

    Rota, and V

    Seibold, K., R. Rota, and V. Savona (2020), Phys. Rev. A 101(3), 033839. Seifert, U. (2012), Rep. Prog. Phys75(12), 126001. Serbyn, M., D. A. Abanin, and Z. Papi´ c (2021), Nat. Phys. 17(6),

    2020

  39. [276]

    Zens, M., D. O. Krimer, and S. Rotter (2019), Phys. Rev. A 100, 013856. Zhang, J.-S. (2020), Int. J. Quantum Inf.18(03), 2050005. Zhang, L., A. E. Motter, and T. Nishikawa (2017), Phys. Rev. Lett.118, 174102. Zhang, L., Z. Wang, Y. Wang, J. Zhang, Z. Wu, J. Jie, and Y. Lu (2023), Phys. Rev. Res.5, 033209. Zhang, M., G. S. Wiederhecker, S. Manipatruni, A. ...

    2019

  40. [285]

    Clerk, A. A., M. H. Devoret, S. M. Girvin, F. Marquardt, and R. J. Schoelkopf (2010), Rev. Mod. Phys.82,

    2010

  41. [293]

    Wittkowski (2023), Phys

    Evers, M., and R. Wittkowski (2023), Phys. Scr.98(12), 125240. Fazio, R., J. Keeling, L. Mazza, and M. Schir` o (2025), SciPost Phys. Lect. Notes ,

    2023

  42. [299]

    Eghbali-Arani, A

    Ameri, V., M. Eghbali-Arani, A. Mari, A. Farace, F. Kheiran- dish, V. Giovannetti, and R. Fazio (2015), Phys. Rev. A 91, 012301. Amitai, E., N. L¨ orch, A. Nunnenkamp, S. Walter, and C. Bruder (2017), Phys. Rev. A95, 053858. Andronov, A. A., A. A. Vitt, and S. E. Khaikin (1966), Theory of Oscillators(Pergamon Press, Oxford, UK). Antonelli, P., and P. Marc...

    2015

  43. [336]

    Wadenpfuhl, K., and C. S. Adams (2023), Phys. Rev. Lett. 131(14), 143002. Walls, D. F., and G. J. Milburn (2008),Quantum optics (Springer Science & Business Media). Walter, S., A. Nunnenkamp, and C. Bruder (2014), Phys. Rev. Lett.112, 094102. 49 Walter, S., A. Nunnenkamp, and C. Bruder (2015), Ann. Phys. (Berlin)527(1-2),

    2023

  44. [339]

    M., and Y

    Hriscu, A. M., and Y. V. Nazarov (2013), Phys. Rev. Lett. 110, 097002. Hush, M. R., W. Li, S. Genway, I. Lesanovsky, and A. D. Armour (2015), Phys. Rev. A91, 061401. H¨ ohe, F., L. Danner, C. Padurariu, B. I. C. Donvil, J. Anker- hold, and B. Kubala (2025), New J. Phys.27(2), 023039. Iemini, F., D. Chang, and J. Marino (2024), Phys. Rev. A 109, 032204. Ie...

    2013

  45. [351]

    Chaparro, D

    Agarwal, S., E. Chaparro, D. Barberena, A. P. n. Orioli, G. Ferioli, S. Pancaldi, I. Ferrier-Barbut, A. Browaeys, and A. Rey (2024), PRX Quantum5, 040335. Aifer, M., J. Thingna, and S. Deffner (2024), Phys. Rev. Lett. 133, 020401. Alaeian, H., and B. Buˇ ca (2022), Commun. Phys.5(1),

    2024

  46. [358]

    Morrell, M. C., L. Elliott, and D. G. Grier (2026), Phys. Rev. Lett.136(5). Mukherjee, A., Y. Ibrahim, M. Hajduˇ sek, and S. Vinjanam- pathy (2024), Phys. Rev. A110(1), 012220. Murch, K. W., S. Weber, C. Macklin, and I. Siddiqi (2013), Nature502(7470),

    2026

  47. [363]

    Luca Giorgi, and R

    Galve, F., G. Luca Giorgi, and R. Zambrini (2017), inQuan- tum Science and Technology(Springer International Pub- lishing, Cham) pp. 393–420. Gambuzza, L. V., A. Cardillo, A. Fiasconaro, L. Fortuna, J. G´ omez-Garde˜ nes, and M. Frasca (2013), Chaos23(4), 043103. Gardiner, C., and P. Zoller (2004),Quantum noise: a hand- book of Markovian and non-Markovian...

    2017

  48. [368]

    Karpat, G., i. d. I. Yal¸ c ınkaya, and B. C ¸ akmak (2019), Phys. Rev. A100, 012133. Karpat, G. b. u., i. d. I. Yal¸ cinkaya, B. i. e. i. f. m. c. C ¸ akmak, G. L. Giorgi, and R. Zambrini (2021), Phys. Rev. A103, 062217. Kato, Y., and H. Nakao (2021a), New J. Phys.23(1), 013007. Kato, Y., and H. Nakao (2021b), Phys. Rev. Res.3, 013085. Kato, Y., and H. N...

    2019

  49. [369]

    Chan, C.-K., T. E. Lee, and S. Gopalakrishnan (2015), Phys. Rev. A91, 051601. Chen, Y.-H., and X. Zhang (2023), Nat. Commun.14(1),

    2015

  50. [380]

    Siddique, A. B., L. Pecora, J. D. Hart, and F. Sorrentino (2018), Phys. Rev. E97, 042217. Singh, V., E. Kwon, and G. J. Milburn (2025), arXiv:2503.12118 [quant-ph]. Siwiak-Jaszek, S., and A. Olaya-Castro (2019), Faraday Dis- cuss.216,

    Pith/arXiv arXiv 2018

  51. [403]

    Roberts, D., and A. A. Clerk (2023), Phys. Rev. Lett.131, 190403. Rohden, M., A. Sorge, M. Timme, and D. Witthaut (2012), Phys. Rev. Lett.109, 064101. Rotondo, P., M. Marcuzzi, J. P. Garrahan, I. Lesanovsky, and M. M¨ uller (2018), J. Phys. A51(11), 115301. Roulet, A., and C. Bruder (2018a), Phys. Rev. Lett.121, 063601. Roulet, A., and C. Bruder (2018b), ...

    2023

  52. [411]

    Blais, A., A. L. Grimsmo, S. M. Girvin, and A. Wallraff (2021), Rev. Mod. Phys.93, 025005. Boccaletti, S., J. Kurths, G. Osipov, D. Valladares, and C. Zhou (2002), Phys. Rep.366(1-2),

    2021

  53. [422]

    (2003),Chemical Oscillations, Waves, and Turbulence(Springer Berlin, Heidelberg)

    Kuramoto, Y. (2003),Chemical Oscillations, Waves, and Turbulence(Springer Berlin, Heidelberg). Lai, D.-G., A. Miranowicz, and F. Nori (2025), Nat. Com- mun.16(1),

    2003

  54. [445]

    Fowler-Wright, P., K. B. Arnard´ ottir, P. Kirton, B. W. Lovett, and J. Keeling (2023), Phys. Rev. Res.5, 033148. Fruchart, M., R. Hanai, P. B. Littlewood, and V. Vitelli (2021), Nature592(7854),

    2023

  55. [459]

    Parlitz (1995), Phys

    Kocarev, L., and U. Parlitz (1995), Phys. Rev. Lett.74,

    1995

  56. [473]

    Lohe, M. A. (2010), J. Phys. A43(46), 465301. L¨ orch, N., S. E. Nigg, A. Nunnenkamp, R. P. Tiwari, and C. Bruder (2017), Phys. Rev. Lett.118, 243602. Lorenzo, S., B. Militello, A. Napoli, R. Zambrini, and G. M. Palma (2022), New J. Phys.24(2), 023030. Louren¸ co, A. C., L. F. dos Prazeres, T. O. Maciel, F. Iemini, and E. I. Duzzioni (2022), Phys. Rev. B1...

    2010

  57. [491]

    Kurths (1997), Phys

    Voss, H., and J. Kurths (1997), Phys. Lett. A234(5),

    1997

  58. [545]

    Huber, R

    Viotti, L., M. Huber, R. Fazio, and G. Manzano (2026), Phys. Rev. Lett.136, 110401. Volovik, G. E. (2013), JETP Letters98(8),

    2026

  59. [547]

    Bruder (2025), Phys

    Kehrer, T., and C. Bruder (2025), Phys. Rev. A112, 012223. Kehrer, T., C. Bruder, and P. Solanki (2025), Phys. Rev. Lett.135, 063601. Kehrer, T., T. Nadolny, and C. Bruder (2024), Phys. Rev. A 110, 042203. Kessler, E. M., G. Giedke, A. Imamoglu, S. F. Yelin, M. D. Lukin, and J. I. Cirac (2012), Phys. Rev. A86, 012116. Keßler, H., J. G. Cosme, M. Hemmerlin...

    2025

  60. [563]

    Frasca, G

    Bergner, A., M. Frasca, G. Sciuto, A. Buscarino, E. J. Ngamga, L. Fortuna, and J. Kurths (2012), Phys. Rev. E85, 026208. Bhaseen, M. J., J. Mayoh, B. D. Simons, and J. Keeling (2012), Phys. Rev. A85, 013817. Bier, M., B. M. Bakker, and H. V. Westerhoff (2000), Bio- phys. J.78(3),

    2012

  61. [587]

    (1986), Physics Today39(12),

    Bak, P. (1986), Physics Today39(12),

    1986

  62. [631]

    Sanpera, R

    Gribben, D., A. Sanpera, R. Fazio, J. Marino, and F. Iemini (2025), SciPost Phys.18(3),

    2025

  63. [642]

    M., and G

    Wiseman, H. M., and G. J. Milburn (2009),Quantum mea- surement and control(Cambridge university press). Witthaut, D., S. Wimberger, R. Burioni, and M. Timme (2017), Nat. Commun.8(1), 14829. Wu, X., Z. Wang, F. Yang, R. Gao, C. Liang, M. K. Tey, X. Li, T. Pohl, and L. You (2024), Nat. Phys.20(9),

    2009

  64. [670]

    Bruder, and A

    Koppenh¨ ofer, M., C. Bruder, and A. Roulet (2020), Phys. Rev. Res.2, 023026. Koppenh¨ ofer, M., and A. Roulet (2019), Phys. Rev. A99, 043804. Kreuz, T. (2013), Principles of Neural Coding ,

    2020

  65. [675]

    C., and A

    Sethia, G. C., and A. Sen (2014), Phys. Rev. Lett.112, 144101. Setoyama, W., and Y. Hasegawa (2024), Phys. Rev. Lett. 132, 093602. Setoyama, W., and Y. Hasegawa (2025), Phys. Rev. A111, 012202. Settimo, F., and B. Vacchini (2026), Phys. Rev. A113, 022213. Shankar, A., J. Cooper, J. G. Bohnet, J. J. Bollinger, and M. Holland (2017), Phys. Rev. A95, 033423....

    Pith/arXiv arXiv 2014

  66. [684]

    WE-Heraeus-Seminar, Bad Honnef, Germany, 2–5 December 2018(Springer) pp. 73–89. Giorgi, G. L., F. Galve, G. Manzano, P. Colet, and R. Zam- brini (2012), Phys. Rev. A85, 052101. Giorgi, G. L., F. Galve, and R. Zambrini (2016), Phys. Rev. A94, 052121. Giorgi, G. L., F. Plastina, G. Francica, and R. Zambrini (2013), Phys. Rev. A88, 042115. Girolami, D., T. T...

    2018

  67. [731]

    Zaletel, M. P., M. Lukin, C. Monroe, C. Nayak, F. Wilczek, and N. Y. Yao (2023), Rev. Mod. Phys.95, 031001. Zanette, D. H., and A. S. Mikhailov (1998), Phys. Rev. E 57,

    2023

  68. [763]

    Passarelli, P

    Postavov´ a, E., G. Passarelli, P. Lucignano, and A. Russo- manno (2026), New J. Phys.28(3), 034502. dos Prazeres, L. F., L. d. S. Souza, and F. Iemini (2021), Phys. Rev. B103(18), 184308. Puri, R. R., and S. V. Lawande (1979), Phys. Lett. A72(3),

    2026

  69. [783]

    Ritsch (2015), Phys

    Piazza, F., and H. Ritsch (2015), Phys. Rev. Lett.115(16), 163601. Piccitto, G., M. Wauters, F. Nori, and N. Shammah (2021), Phys. Rev. B104(1), 014307. Pikovsky, A., M. Rosenblum, and J. Kurths (2001), Cam- bridge university press12. Plenio, M. B., and P. L. Knight (1998), Rev. Mod. Phys.70,

    2015

  70. [821]

    Casado-Pascual, M

    Goychuk, I., J. Casado-Pascual, M. Morillo, J. Lehmann, and P. H¨ anggi (2006), Phys. Rev. Lett.97, 210601. Greilich, A., N. E. Kopteva, A. N. Kamenskii, P. S. Sokolov, V. L. Korenev, and M. Bayer (2024), Nat. Phys.20(4),

    2006

  71. [885]

    Schmolke, C.-K

    Tao, Z., F. Schmolke, C.-K. Hu, W. Huang, Y. Zhou, J. Zhang, J. Chu, L. Zhang, X. Sun, Z. Guo,et al.(2025), Nat. Commun.16(1),

    2025

  72. [932]

    Lutz (2022), Phys

    Schmolke, F., and E. Lutz (2022), Phys. Rev. Lett.129, 250601. Schmolke, F., and E. Lutz (2024), Phys. Rev. Lett.132, 010402. Schmolke, F., and E. Lutz (2026), arXiv:2606.21226 [quant- ph]. Schnell, A., A. Eckardt, and S. Denisov (2020), Phys. Rev. B101, 100301. Schumann, J. C., I. Lesanovsky, and P. Solanki (2026), arXiv:2601.09779 [quant-ph]. Scovil, H....

    Pith/arXiv arXiv 2022

  73. [978]

    van der Mark (1928), London Edin- burgh Dublin Philos

    van der Pol, B., and J. van der Mark (1928), London Edin- burgh Dublin Philos. Mag. J. Sci.6,

    1928

  74. [980]

    Pohl (2025), arXiv:2503.16141 [quant-ph]

    Russo, F., and T. Pohl (2025), arXiv:2503.16141 [quant-ph]. Sacha, K. (2015), Phys. Rev. A91, 033617. Sacha, K., and J. Zakrzewski (2017), Rep. Prog. Phys.81(1), 016401. Sachdev, S. (2000),Quantum Phase Transitions(Cambridge University Press). Sarfati, R., J. C. Hayes, and O. Peleg (2021), Sci. Adv.7(28), eabg9259. Scarlatella, O., A. A. Clerk, R. Fazio, ...

    Pith/arXiv arXiv 2025

  75. [986]

    Raskatla, J

    Liu, T., V. Raskatla, J. Li, K. F. MacDonald, and N. I. Zheludev (2025b), Newton1(7), 100206. Lled´ o, C., Th. K. Mavrogordatos, and M. H. Szyma´ nska (2019), Phys. Rev. B100(5), 054303. Lled´ o, C., and M. H. Szyma´ nska (2020), New J. Phys.22(7), 075002. Lodahl, P., S. Mahmoodian, S. Stobbe, A. Rauschenbeutel, P. Schneeweiss, J. Volz, H. Pichler, and P....

    2019

  76. [995]

    Fazio, and C

    Casteels, W., R. Fazio, and C. Ciuti (2017), Phys. Rev. A 95, 012128. Cattaneo, M., G. L. Giorgi, S. Maniscalco, G. S. Paraoanu, and R. Zambrini (2021), Ann. Phys.533(5), 2100038. Cebri´ an-Lacasa, D., P. Parra-Rivas, D. Ruiz-Reyn´ es, and L. Gelens (2024), Phys. Rep.1096,

    2017

  77. [1087]

    Binder, F., L. A. Correa, C. Gogolin, J. Anders, and G. Adesso (2018), Fundamental Theories of Physics 195(1). Blair, K. (1915), Nature96(2406),

    2018

  78. [1100]

    Majhi, B

    Kundu, S., S. Majhi, B. K. Bera, D. Ghosh, and M. Laksh- manan (2018), Phys. Rev. E97, 022201. Kuramoto, Y. (1975), inInternational Symposium on Mathe- matical Problems in Theoretical Physics, edited by H. Araki (Springer Berlin Heidelberg, Berlin, Heidelberg) pp. 420–

    2018

  79. [1123]

    V., and D

    Zhirov, O. V., and D. L. Shepelyansky (2008), Phys. Rev. Lett.100, 014101. Zhu, B., J. Schachenmayer, M. Xu, F. Herrera, J. G. Restrepo, M. J. Holland, and A. M. Rey (2015), New J. Phys.17(8), 083063

    2008

  80. [1155]

    Coombes, S., and P. C. Bressloff (1999), Phys. Rev. E60,

    1999

Showing first 80 references.