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Bistability and nonequilibrium condensation in a driven-dissipative Josephson array: a c-field model

by Matthew T. Reeves, Matthew J. Davis

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Submission summary

Authors (as registered SciPost users): Matthew Davis · Matt Reeves
Submission information
Preprint Link:  (pdf)
Date accepted: 2023-05-30
Date submitted: 2023-05-05 07:49
Submitted by: Reeves, Matt
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • Atomic, Molecular and Optical Physics - Theory
Approach: Computational


Developing theoretical models for nonequilibrium quantum systems poses significant challenges. Here we develop and study a multimode model of a driven-dissipative Josephson junction chain of atomic Bose-Einstein condensates, as realised in the experiment of Labouvie et al. [Phys. Rev. Lett. 116, 235302 (2016)]. The model is based on c-field theory, a beyond-mean-field approach to Bose-Einstein condensates that incorporates fluctuations due to finite temperature and dissipation. We find the c-field model is capable of capturing all key features of the nonequilibrium phase diagram, including bistability and a critical slowing down in the lower branch of the bistable region. Our model is closely related to the so-called Lugiato-Lefever equation, and thus establishes new connections between nonequilibrium dynamics of ultracold atoms with nonlinear optics, exciton-polariton superfluids, and driven damped sine-Gordon systems.

Published as SciPost Phys. 15, 068 (2023)

Author comments upon resubmission

Dear Editor,

We are glad to see that the reviewers feel the manuscript struck the right balance between modelling the specific experiment and a general theoretical development, as was our aim for this work. We thank the reviewers for their comprehensive and broadly positive assessment of our work, and their helpful comments/suggestions.

We have made changes to address all of the minor points raised by the reviewers, as detailed below. We believe the manuscript is now suitable for publication.

List of changes

Changes in Response to Reviewer 1

1. The number of single-particle states that need to be considered is determined by the chemical potential. So, yes --- equivalently, one could use the Thomas Fermi radius, since they are related via the relation R = \ sqrt(2 mu/ m omega^2). As detailed in the manuscript, for the problem at hand, a cutoff in the range 2-3 mu was found to be appropriate. To address this point, we have expanded the explanation of the projector after Eq. (4). We have also included additional references, which explain the rationale of choosing the single particle basis states to define the projector / cutoff energy.

2. We have relabelled the "quasi condensate (QC)" state as a “nonequilibrium quasi-condensate (NQC)” throughout the text and all the figures, to help distinguish this nonequilibrium steady state from the traditional (equilibrium) “quasi-condensate”.

3. Expanded the text on pg. 7 explaining how the NQC state emerges from the normal state.

4. Added the parameter values to the caption of Fig. 3. The additional text provided in response to point 3 also gives additional information about the emergence of the non equilibrium quasi-condensate from the normal state.

5. We have termed g_2 as the "second order correlation function" throughout. However, we still refer to density and phase "coherences" when referring to g_1 and g_2 respectively, as this is physically what these correlation functions measure.

6. No change (the figure labels are correct). In the region bounded by the blue circles and the orange stars, the system is in the SF state when initialized on the full branch (a), and is in the normal state when initialized in the empty branch (b). This can be seen by comparing the values of the condensate fraction presented in Figs 5a and 5b within the region bounded by those two boundaries.

7. For our purposes the effective temperature is simply a phenomenological fitting parameter; it incorporates additional sources of noise (thermal, etc.), but there is no rationale for choosing a particular value besides tuning the phase boundaries. We have added additional text after Eq. (30) explaining this point further.

8. Added the following to the caption of Fig. 10: “The star markers show the observed values where critical slowing down occured in the experiment.” We also added the following text on pg. 10: “For completeness, in Fig. 10 we also show the points where critical slowing down was observed in the experiment [orange stars], although no signature of this boundary appears in our dynamical reservoir model.” As we suggest at the end of the discussion, we suspect that simulating the dynamics of the full chain may be needed to capture this boundary correctly.

Changes in Response to Reviewer 2:

1. Corrected typographical error

2. Added the following text after Eq. (1): “L is the Gross-Pitaevskii operator, γ is the particle loss rate, F is the driving associated with the reservoir sites, and dW is a Wiener noise term associated with the losses.” We then go into the specifics of each term.

3. Corrected typographical error

4. We have changed the text to read “Eq. (18) Permits either one or three solutions (real and positive) …”. We also agree that some of the explanation in this section was probably not required; in the interest of brevity, we have therefore also shortened the explanation throughout this section by removing several remarks that were not essential.

5. We have changed the text to read “differentiating Eq. (18) with respect to n_S yields …”

6. We have ensured the x and y ticks in Fig. 2(d) are the same size and that the figures have consistent fonts throughout the manuscript.

7. We have added the following text in Sec. IV, pg 5: “To verify that our results did not depend on the choice of cutoff, we calculated the condensate number $N_0$, as determined from the Onsager-Penrose criterion [31], which specifies the condensate number as the largest eigenvalue of the one-body density matrix”.

8. We agree that the term “instantaneous chemical potential” is a loose concept and this interpretation is only valid under certain conditions and assumptions. We have rephrased the text after Eq. (32) as follows: “The quantity $\bar \mu$ may be loosely interpreted as an “instantaneous chemical potential”, although strictly the chemical potential is only defined in equilibrium.” Also, there is no summation convention; for clarity we have therefore added the subscript $j$ to the quantity $\bar \mu$ in Eq. (32).

9. We think this is an interesting observation and have added the following text in the relevant paragraph: “Similar behaviour is well known in the study of superfluid helium through capillaries, wherein the superfluid can easily flow but the normal fluid cannot.”

10. We have changed the text to the following: “the system can still be reduced to a single mode in terms of the stationary states of the nonlinear GPE operator $\mathcal{L}$”

Reports on this Submission

Anonymous Report 2 on 2023-5-9 (Invited Report)


The authors have responded very creditably to the comments of both referees, and I am happy to approve the manuscript as it stands.

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Anonymous Report 1 on 2023-5-5 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:2102.02949v3, delivered 2023-05-05, doi: 10.21468/SciPost.Report.7149


As I pointed out in my previous report, the manuscript provides a very well written and carefully conducted theoretical and numerical study of an intriguing experiment. While it still not fully explains every aspect of the experiment, it provides new insights into possible mechanisms explaining the experimental data. Since, moreover, the authors addressed the (minor) points raised in my previous report in a satisfactory fashion, I am happy to recommend the publication of the revised manuscript in SciPost Physics in its present form.

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  • significance: -
  • originality: -
  • clarity: -
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