# Algebraic Theory of Quantum Synchronization and Limit Cycles under Dissipation

### Submission summary

 As Contributors: Cameron Booker · Berislav Buca Arxiv Link: https://arxiv.org/abs/2103.01808v4 (pdf) Date submitted: 2021-11-12 04:22 Submitted by: Buca, Berislav Submitted to: SciPost Physics Academic field: Physics Specialties: Atomic, Molecular and Optical Physics - Theory Quantum Physics Statistical and Soft Matter Physics Approach: Theoretical

### Abstract

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.

###### Current status:
Has been resubmitted

Dear Editor,

We would like to thank you and the referees for carefully assessing our work. As requested, we have revised our manuscript 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 improvements acceptable.

However, we ask you to reconsider your recommendation to transfer our work to SciPost Physics Core. All three referees' reports stated that the work was suitable for publication in SciPost Physics subject to revisions. The referees also stated that the work was of 'high' significance. The only report raising criticism with the results and not the presentation is No. 3. They refer to the scope and novelty of our work, e.g. claiming that we can only treat identical subsystems and weak perturbations from that. This is incorrect, and we now give an explicit example with subsystems that are non-perturbatively different. The rest of the criticism stems from differences in terminology, which we have now fully addressed. This further demonstrates the importance of linking different areas of quantum synchronization together, which we provide as noted by Ref. 2, and aligns with one of the acceptance criteria of SciPost Physics. Our results are generally applicable to time-independent quantum master equations as stated before and are entirely novel because they give both the necessary and sufficient conditions for quantum limit cycles and (spontaneous) synchronization. As outlined in our cover letter, our fundamental theoretical results provide a paradigm shift linking various fields of quantum synchronization and non-linear dynamics together. This opens a new research direction, with several follow up works already proposed as evidenced by the preprint accumulating 10 citations since being posted on the arXiv in March: https://ui.adsabs.harvard.edu/abs/2021arXiv210301808B/abstract. Thus we believe that we have firmly met the criteria for publication in SciPost Physics.

Below we reply to each referee individually.

Yours sincerely,
Berislav Buca, Cameron Booker and Dieter Jaksch

### List of changes

Note that we have not listed minor corrections of spelling, grammar and punctuation.
Abstract:
Clarify that we are studying spontaneous synchronisation in time-homogeneous systems.

Section 1
1. The sentence Indeed, even when the stationary state can be found analytically diagonalizing it, as is required to find its support, is still an open problem.'' on page 8 has been replaced with Although in some cases analytic tools can be used to find non-equilibrium steady states, it remains an open problem to efficiently find the support of a generic stationary state.''
2. Rewording of introduction to clarify that we are considering un-driven spontaneous synchronization.
3. Streamlining' of Sec. 1.1.1 to only focus on topics relevant to our study. References to Haken's work on Synergetisc have also been included.

Section 2
1. The remark on page 9 regarding exponentially small terms has been made into its own paragraph.
2. On page 11 the phrase trivial Jordan form'' has been replaced with diagonalizable''.
3. Added a footnote to explain that what we call meta-stable synchronization has also been called transient synchronization in the past.
4. Modification to the introduction of the Lindblad formalism to make clearer how our theory can be applied to both markovian and non-markovian systems.
5. Clarified that our discussion refers to synchronization between identical subsystems.

Section 3
1. The manipulation to show the non-unitarity of $A$ in Thm. 2 has been added.
2. The discussion regarding commensurability has been made into its own subsection and reworded to be clearer.
3. Added an example to indicate the modifications required in order to apply our to phase synchronization.

Section 4
1. Subsection 4.3 we emphasise that $\omega \ne 0$. We also now emphasize the distinction between the cases in 4.1 and 4.3.
2. Change the sentence after Thm. 3 to Crucially, this means that at first order in $s$ the perturbed eigenvalues remain purely imaginary.''
3. In Cor. 4 we insert the term anti-symmetry'' to make it clearer what sort of symmetry breaking we are considering.

Section 5
1. The discussion regarding constructing more general models exhibiting quantum synchronisation has been moved to the appendices.

Section 6
1. Section 6 has been moved to before discussion of various examples. We have also included an example of when this result can be used following the work of Prosen in Ref. [169].

Appendices
1. The discussion regarding constructing more general models exhibiting quantum synchronisation has been moved to the appendices.
2. A new appendix has been added to explain how our theory can be applied to non-Markovian environments.

### Submission & Refereeing History

#### Published as SciPost Phys. 12, 097 (2022)

Resubmission 2103.01808v5 on 12 January 2022

Resubmission 2103.01808v4 on 12 November 2021
Submission 2103.01808v2 on 3 May 2021

## Reports on this Submission

### Anonymous Report 3 on 2021-12-10 (Invited Report)

• Cite as: Anonymous, Report on arXiv:2103.01808v4, delivered 2021-12-10, doi: 10.21468/SciPost.Report.4024

### Report

I acknowledge the authors for their revisions and answers. Focusing on the manuscript main content, and beyond presentation refinements, I appreciate the formal presentation of this manuscript and additions to the previous one, but I have to insist on two points.

The first is about the claimed impossibility to synchronize two ½ spins. The authors argument stands on (their) unusual definition of synchronization for which two temporal signals:
cos(t)
a+cos(t)
would not be synchronized, due to the presence of an offset.
Generally synchronization is considered looking at the dynamical part of signals and any constant off-set is actually neglected. Beyond the authors choice of “terminology”, they should notice that actually these signals would be perfectly synchronized also using the same synchronization indicator of their Eqs. 1 and 2.

The second is about the applicability of this framework to non-identical subsystems.
“The claim that we cannot treat subsystems that are not identical is incorrect. We now give an explicit example in Sec. 6.2.1 ...”
Maybe there was a misunderstanding, as I was referring specifically to synchronization.
Indeed “stable quantum synchronization” is addressed in Section 3.2 providing sufficient conditions in two corollaries. Both of them assume the exchange/permutation symmetry between sites, the first in the Liouvillian and the second in the operator A, rather strong symmetries indeed at the basis of my comment.
The authors do not refer to this in their answer but highlight the example in Sec. 6.2.1, actually already in the previous version of the manuscript. This is a rather elaborated model and in the present form it does not clarify this point.
It is clear that the global S^+ and S^- are permutation invariant, while the Hamiltonian is not.
On the other hand, in order to appreciate the claimed generality, could the authors specify the local operators Oj that would be identically synchronized in this inhomogeneous model?
It would be also useful to include the definition of c_j in the “agnostic” local ladders (38) and (39).

As a note, the newly added theorem 4 would be really less obscure clarifying the meaning of the “commutant” proportional to the Identity.

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

### Author:  Berislav Buca  on 2022-01-12  [id 2095]

(in reply to Report 3 on 2021-12-10)
Category:
correction

We thank the referee for their careful consideration of our work and detailed comments. Here we respond to their requested changes directly and outline how we have improved the manuscript accordingly.

Point 1:

"The first is about the claimed impossibility to synchronize two ½ spins. The authors argument stands on (their) unusual definition of synchronization for which two temporal signals:
\begin{equation*}
\end{equation*}
would not be synchronized, due to the presence of an offset.
Generally synchronization is considered looking at the dynamical part of signals and any constant off-set is actually neglected. Beyond the authors choice of “terminology”, they should notice that actually these signals would be perfectly synchronized also using the same synchronization indicator of their Eqs. 1 and 2."

Firstly, we must emphasise that we have nowhere claimed that it is impossible synchronize two spin-1/2s. In our response to the referee's previous comments, we provided an example of two spin-1/2s under a common Markovian bath and explained that under our stricter definition, these two spins do not anti-synchronize but that \emph{`under a weaker definition of synchronization we could consider this system to be (anti-)synchronized''.

We chose to work with the strictest definition of synchronization in our work to avoid additional technicalities in the main theorems as these technicalities offer little additional insight. In our previous revision we already made indication of where additional considerations could be made to obtain weaker notions of synchronization. In our latest version we have included a new subsection dedicated to this discussion in order to make these ideas clearer. We have also explicitly pointed out to the reader that our definitions are stricter that the Pearson correlation indicator.

Point 2:
“The claim that we cannot treat subsystems that are not identical is incorrect. We now give an explicit example in Sec. 6.2.1 ...”
Maybe there was a misunderstanding, as I was referring specifically to synchronization.
Indeed “stable quantum synchronization” is addressed in Section 3.2 providing sufficient conditions in two corollaries. Both of them assume the exchange/permutation symmetry between sites, the first in the Liouvillian and the second in the operator A, rather strong symmetries indeed at the basis of my comment.
The authors do not refer to this in their answer but highlight the example in Sec. 6.2.1, actually already in the previous version of the manuscript. This is a rather elaborated model and in the present form it does not clarify this point.
It is clear that the global $S^+$ and $S^-$ are permutation invariant, while the Hamiltonian is not.
On the other hand, in order to appreciate the claimed generality, could the authors specify the local operators $O_j$ that would be identically synchronized in this inhomogeneous model?
It would be also useful to include the definition of $c_j$ in the “agnostic” local ladders (38) and (39)."

We now recognise our misunderstanding, and apologise to the referee. Our framework can only treat systems where the local Hilbert spaces of each site have the same dimension - i.e. we cannot consider synchronizing a spin-1/2 with a spin-1. We have made this clearer in our revised manuscript, and explicitly pointed out that our definition for robustness would no longer be applicable for sites with different local Hilbert space dimensions. We have also outlined in our conclusions that extending our theory for studying synchronisation between subsystems with different local Hilbert spaces is an active focus of future work.

We now recognise our misunderstanding, and apologise to the referee. Our framework can only treat systems where the local Hilbert spaces of each site have the same dimension - i.e. we cannot consider synchronizing a spin-1/2 with a spin-1. We have made this clearer in our revised manuscript, and explicitly pointed out that our definition for robustness would no longer be applicable for sites with different local Hilbert space dimensions. We have also outlined in our conclusions that extending our theory for studying synchronisation between subsystems with different local Hilbert spaces is an active focus of future work.

Finally, we thank the referee for identifying a typographical mistake, Eqs. (38) and (38) should read
\begin{align*}
L^{-}_{j}&=\gamma^-_{j} c_{j, \uparrow}c_{j, \downarrow}\\
L^+_{j}&=\gamma^+_{j} c^\dagger_{j, \uparrow}c^\dagger_{j, \downarrow}
\end{align*}

Point 3:
"As a note, the newly added theorem 4 would be really less obscure clarifying the meaning of the “commutant” proportional to the Identity."

This has been clarified by a footnote.

### Anonymous Report 1 on 2021-12-5 (Invited Report)

• Cite as: Anonymous, Report on arXiv:2103.01808v4, delivered 2021-12-05, doi: 10.21468/SciPost.Report.3946

### Report

The presentation of the manuscript has significantly improved in this revision - the structure is now much cleaner, the introduction sharper and the presentation of the main theorems and proofs is more approachable to a non-specialised reader.

In particular, the authors refined their introduction to the Lindblad equation. Nevertheless, I would appreciate if the authors decided for a uniform approach as to whether they focus on the Markovian "spirit" of the master equation or whether they want to emphasise non-Markovian extensions. The authors argue that one may place part of the environment inside the system's Hilbert space to allow for non-Markovian dynamics. However, one might argue that the actual environment represented by the Lindblad operators is still purely Markovian. Note that there also exist alternative extensions to bring the Lindblad formalism to the non-Markovian regime, see e.g. the work by Breuer (Phys. Rev. A 75, 022103) or the work by Head-Marsden and Maziotti (Phys. Rev. A 99, 022109), but in my view these are all extensions to the - in itself Markovian - Lindblad formalism. Although my concern is primarily semantic, I think it is nonetheless worthwhile to stick as close as possible to the established literature when it comes to discussing the Lindblad master equation framework.
In any case, the discussion in Appendix K is much appreciated. Nevertheless, I think it would be important to include the central conclusion "[...] it is not possible to draw general conclusions directly about the dynamics of the system without knowing the details of the non-Markovian bath and applying our results to the combined system." in the main text.

Finally, I would recommend another thorough proofreading of the text to amend the remaining grammatical issues. For example, in the introduction it should be "characterize" in the sentence "These results provide a general theory for studying quantum synchronization and characterizes the phenomenon." Another example is at the end of the introductory text for 4.1 where I suppose that by "We also remark that generically there is only one order of s different between the decay rate and the oscillation frequency." the authors probably mean "[...] the order of s only differs by one between the decay rate [...]", i.e., it is not about the existence of a different order but rather that the order itself is different, if I understand it well.

In conclusion, barring the two minor remarks mentioned above, I can now recommend this manuscript for publication in SciPost Physics.

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

### Author:  Berislav Buca  on 2022-01-12  [id 2094]

(in reply to Report 1 on 2021-12-05)
Category:
correction

We thank the referee for their careful consideration of our work and detailed comments. Here we respond to their requested changes directly and outline how we have improved the manuscript accordingly.

Point 1:
"The authors refined their introduction to the Lindblad equation. Nevertheless, I would appreciate if the authors decided for a uniform approach as to whether they focus on the Markovian "spirit" of the master equation or whether they want to emphasise non-Markovian extensions. The authors argue that one may place part of the environment inside the system's Hilbert space to allow for non-Markovian dynamics. However, one might argue that the actual environment represented by the Lindblad operators is still purely Markovian. Note that there also exist alternative extensions to bring the Lindblad formalism to the non-Markovian regime, see e.g. the work by Breuer (Phys. Rev. A 75, 022103) or the work by Head-Marsden and Maziotti (Phys. Rev. A 99, 022109), but in my view these are all extensions to the - in itself Markovian - Lindblad formalism. Although my concern is primarily semantic, I think it is nonetheless worthwhile to stick as close as possible to the established literature when it comes to discussing the Lindblad master equation framework.
In any case, the discussion in Appendix K is much appreciated. Nevertheless, I think it would be important to include the central conclusion "[...] it is not possible to draw general conclusions directly about the dynamics of the system without knowing the details of the non-Markovian bath and applying our results to the combined system." in the main text."

As advised by the referee, we have now restricted the discussion in the main text so that it focuses exclusively on the Markovian regime. We have then amended our appendix slightly and included the references suggested by the referee.

Point 2:
"Finally, I would recommend another thorough proofreading of the text to amend the remaining grammatical issues. For example, in the introduction it should be "characterize" in the sentence "These results provide a general theory for studying quantum synchronization and characterizes the phenomenon." Another example is at the end of the introductory text for 4.1 where I suppose that by "We also remark that generically there is only one order of s different between the decay rate and the oscillation frequency." the authors probably mean "[...] the order of s only differs by one between the decay rate [...]", i.e., it is not about the existence of a different order but rather that the order itself is different, if I understand it well."

We thank the referee for their careful reading of our manuscript. As advised, we have thoroughly proof read the manuscript again and eliminated the remaining grammatical issues found.

### Strengths

see my first report

### Weaknesses

see my first report

### Report

see my first report

### Requested changes

see my first report

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