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Algebraic Theory of Quantum Synchronization and Limit Cycles under Dissipation

by Berislav Buca, Cameron Booker, Dieter Jaksch

This Submission thread is now published as

Submission summary

Authors (as registered SciPost users): Cameron Booker · Berislav Buca · Dieter Jaksch
Submission information
Preprint Link:  (pdf)
Date accepted: 2022-02-16
Date submitted: 2022-01-12 16:53
Submitted by: Buca, Berislav
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • Atomic, Molecular and Optical Physics - Theory
  • Quantum Physics
  • Statistical and Soft Matter Physics
Approach: Theoretical


Synchronization is a phenomenon where interacting particles lock their motion and display non-trivial dynamics. Despite intense efforts studying synchronization in systems without clear classical limits, no comprehensive theory has been found. We develop such a general theory based on novel necessary and sufficient algebraic criteria for persistently oscillating eigenmodes (limit cycles) of time-independent quantum master equations. We show these eigenmodes must be quantum coherent and give an exact analytical solution for all such dynamics in terms of a dynamical symmetry algebra. Using our theory, we study both stable synchronization and metastable/transient synchronization. We use our theory to fully characterise spontaneous synchronization of autonomous systems. Moreover, we give compact algebraic criteria that may be used to prove absence of synchronization. We demonstrate synchronization in several systems relevant for various fermionic cold atom experiments.

Author comments upon resubmission

Dear Editor,

We would like to thank you and the referees again for carefully assessing our responses and the revised manuscript, and for recommending publication in SciPost Physics subject to minor amendments. We have now made further revisions to our work according to the referees' suggestions, and we have attached a list of changes together with responses to each of the referees' individual comments.

We hope they find our minor amendments acceptable.

Yours sincerely,
Berislav Buca, Cameron Booker and Dieter Jaksch

List of changes

1. In Sec. 2.1 we have explicitly pointed out to the reader that our definitions are stricter that the Pearson correlation indicator.
2. We have added a new subsection 3.4 to discuss in more detail adaptations of our results to weaker notions of synchronization.
3. A footnote has been added to Theorem 4 defining the commutant of a set.
4. References have been added to appendix K as suggested by referee 1.

Published as SciPost Phys. 12, 097 (2022)

Reports on this Submission

Anonymous Report 2 on 2022-1-21 (Invited Report)


In the revised manuscript most points have been clarified, most notably with the addition of sect 3.4. I am recommending this manuscript for SciPost Physics.

  • validity: -
  • significance: -
  • originality: -
  • clarity: -
  • formatting: -
  • grammar: -

Anonymous Report 1 on 2022-1-14 (Invited Report)


I appreciate the authors' further revision to their manuscript. All my concerns and remarks were fully addressed such that I recommend this work for publication in SciPost Physics.

  • validity: high
  • significance: high
  • originality: high
  • clarity: high
  • formatting: excellent
  • grammar: good

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