Space-time first-order correlations of an open Bose Hubbard model with incoherent pump and loss

1- The paper presents a thorough analysis of correlations in the steady state of a driven-dissipative Bose-Hubbard model, combining insights from different analytical approaches with numerics.

2- The paper shows that even though the steady state is at infinite temperature, temporal correlations reveal nontrivial dynamics

1- Given that the steady state is at infinite temperature, the ground state of the closed system is not a well-motivated point of reference. In particular, the steady state is not continuously connected to the ground state in the limit $\gamma_p, \gamma_l \to 0$ at fixed ratio $z$.

2- The comparison to the isolated system at infinite temperature in Fig. 7 shows almost identical behavior. Therefore, it is not clear whether the "features of the spectral function are a clear signature of the non-equilibrium nature of the quantum system" as claimed in the introduction.

The paper studies correlations in the steady state of a driven-dissipative Bose-Hubbard model, using a combination of analytical and numerical approaches. The presentation is clear and the results are interesting. In particular, the nontrivial dependence of correlations and spectral features on hopping and interactions, even though the system is essentially at infinite temperature, is perhaps unexpected.

However, I do not think that the claim that the "features of the spectral function are a clear signature of the non-equilibrium nature of the quantum system" is justified. On the one hand, the comparison to the isolated system at zero temperature seems a bit far-fetched; on the other hand, the comparison to the isolated system at infinite temperature shows almost identical features. Therefore, what the consequences of the system being driven and dissipative are remains somewhat unclear.

Moreover, at the moment I do not see how the paper satisfies the criteria of providing "a novel and synergetic link between different research areas," or of opening "a new pathway in an existing or a new research direction." Therefore, I do not think that the paper is suitable for publication in SciPost Physics, at least not in its present form.

1- In Fig. 5, on the left-hand side there seem to be $L = 8$ different values of momenta, but on the right-hand side the values of $k$ look continuous. Why is that the case? Further, for $N_s = 3$, there should be three white lines in the plot on the left-hand side, but I can see only two.

2- What are the blue, red, and green lines in Fig. 6?

3- What is the exact solution shown in the left panel in Fig. 6? How is $\mu$ determined?

4- For small system sizes as shown in Fig. 5 and the left panel of Fig. 6, the difference between a linear and a quadratic dispersion is really hard to see. Is there a better way to show this?

5- In the captions of some figures (e.g., Fig. 7, 8, 9, 10) parameters are not given in units of $\gamma_l$.

6- Do the right panels of Figs. 5 and 7 show the same data? This is somewhat confusing.

7- What is the meaning of the symbol $\omega_{N - 1, L}$ in Eq. (30)?

8- The discussion in Appendix B is unclear. What is the "problem" defined by Eq. (45)? What are solutions to that problem? What determines the value of the upper bound $d_l$? What happens going from Eq. (47) to (48)?

9- It is not quite clear to me why Appendices C and D are included in the manuscript.

10- What is the parameter $\gamma$ in Eq. (50)?

11- I believe in Eq. (52) there is a typo: it should be $\{1, \dotsc, L \} \setminus \{ i \}$.

12- What are the "superoperators" mentioned in Appendix E?

13- Equations (61) and (62) are supposedly expansions in $U \gg J$, but $J$ does not appear in these equations.

14- The discussion of the fitting procedure in the second paragraph of Sec. F.2 is unclear.

15- The caption of Fig. 8 refers twice to the black solid line, which I find somewhat confusing.

16- I do not think that "one-sited" in Sec. F.3 is a commonly used word. What is meant here? Is this the result for a single site? But this would correspond to $J = 0$ and not to small $J$ as stated in the text.

17- What is the parity that is conserved as stated in Sec. F.5?

18- Where is the approach described in Sec. F.5 used? I suppose it is used to obtain the left panel in Fig. 7, but this could also be obtained by using the same approach as for the right panel, simply by setting $\gamma_l = \gamma_p = 0$ in the time evolution.

19- In Fig. 10, it seems surprising that the results for $L = 2, 4$ agree but are way off for $L = 6$. Why is that? In the same figure, what is the meaning of the error bars?

20- The Heaviside function is denoted by both $\theta$ and $\Theta$.

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- Pedagogical introduction
- High clarity of discussion
- High degree of experimental relevance
- Simple and generic model which is widely studied and of interest to a broad community of researchers
- Rigorous comparison to a number of limiting cases and a direct connection to ground-state results, for which readers may be more familiar.

- Direct connection to experiment could be better reflected in references.
- There is little discussion of why the Green's function is expected to have no interaction dependence.

The authors consider the Bose-Hubbard model with uniform gain and loss in a Lindblad master equation. Numerical analysis of the full system is complemented by analytical results in limiting cases. After the first two introductory sections, the steady state itself is discussed and its simple nature is emphasized. The retarded Greens function are computed and compared to results for ground-states, as well as to unitary evolution from the steady state. It is demonstrated that dissipation alters the low energy excitations. It is also shown that despite the fact that the steady state has no signature of the interactions, the relaxation towards this state has a clear dependence on the interactions.

The paper is highly pedagogical and easy to follow, with an intuitive and comprehensive introduction. The techniques employed are appropriate and I believe the results are of significant value to the community given current experimental interest in relatively small Bose-Hubbard arrays. Specialists will appreciate the details, while non-specialists should also be able to follow the clear exposition. The comparison between the dissipative spectral function and that of the ground-state is particularly interesting and clear, emphasising how dissipation changes the nature of excitations. This should assist the reader to make a connection to more familiar results for closed quantum systems.

Specific Comments:

⁃ Overall, I think the direct connection to experiment could be more strongly emphasized by providing a few more references. There is a sentence “While all these studies concern closed quantum systems, in several experimental situations quantum systems are subjected to external pump and/or losses, or put in contact with some type of bath.” Following this, there is a list of “examples of bosonic open systems”, however, only a few experiments are listed. Given the preceding sentence, I was expecting a list of experiments. This is particularly true for [31]-[40]. A few more experimental references here would be appreciated. Possible examples that are not currently present are experiments with lossy Bose-Hubbard arrays in which a subsystem experiences an effective (though possible coherent) gain due to transport from other parts of the lattice. An example can be found in PRL 116, 235302 (2016) and there are many examples of non-equilibrium Bose-Hubbard experiments in the review AVS Quantum Sci. 3, 039201 (2021). These could be used to add to the broader context of this work.


⁃ There are other models without any interaction dependance in the steady states, but non-trivial fluctuations and decay that depend on interactions. However, to my knowledge these are critical systems with dark states, specifically, absorbing dark states where the steady state is a vacuum. For example, PRA 98, 062117 (2018) and PRL 132 120401 (2024). The manner in which the Bose-Hubbard example here is ‘featureless’ is of course different, being an infinite temperature steady state with a given fugacity. Presumably there are simple counter-examples where the steady state is trivial but there is no signature of interactions in the spectral function? Generically, dephasing leads to infinite temperature states. In those cases, would there be any signatures of interactions in the approach to the steady state? Since the authors emphasise that this aspect is “remarkable”, it would be nice to have some discussion of the broader expectations here. Tangential to this, why is it surprising that the decay rate and oscillation frequency would be renormalized by the interactions?

⁃ Fig 3a: A lot of negative times are shown in the figure. Is there a reason for this? It is a little bit distracting and some space could be saved by reducing this.


⁃ Fig 3b: Are the oscillatory features best shown in the log-plot? Perhaps this could be clarified with an inset? I appreciate that the decay may make it difficult to visualise with a linear scale and the current figure may be as clear as it can be.


⁃ Fig 4. I think the color bar label is missing.

Pending clarifications to my questions and subsequent minor changes where appropriate, I am happy to recommend this for publication in SciPost Physics.

1. Improved connection to experiment

2. Better discussion around why it is remarkable that the spectral function has interaction dependence, despite the absence of interaction effects in the steady state.

Ask for minor revision

1- The steady state of this model was already know to be the infinite temperature state with a given fugacity. The novelty of this paper is the study of the first order correlations. The system is studied both with exact diagonalization (ED) and with the Keldysh field formalism. It is only within the ED approach that new results on the correlations are obtained.

2- The paper is well written and easy to follow. The introduction gives a good overview of the field of open many body systems and the theoretical techniques to deal with them.

1- The problem I have with the manuscript is related to the perspective the results are put in. In the abstract and conclusions, the difference with the correlations in the ground state of the Bose-Hubbard model are emphasised. It is a bit a strange choice to put the emphasis on these differences, because the steady state is rather the infinite temperature one. Actually in Sec. 5.4 the comparison with the infinite temperature closed system is made and a good correspondence is observed. At least, this is what I get from Fig. 7 and the describing text. It is said in the abstract that “the dispersion is quadratic instead of linear at small wave vectors”. But in Fig. 7, the dispersion also looks quadratic to me for the closed system.
I would therefore tend to say that the driving and dissipation has a small effect on the first-order correlations. I agree that this conclusion may be a bit disappointing and that the authors therefore put the emphasis on the difference with the zero-temperature state. But that conclusion would also hold for comparing the infinite and zero temperature closed systems and no one would be surprised by that.
Given the close similarities between the left and right hand panels in Fig. 7, I could imagine that it is possible to develop a perturbative treatment of the driving and dissipation, but it is hardly the task of the referee to suggest new research.

The manuscript by Zündel et al. presents a theoretical analysis of the first order correlation function of a driven-dissipative Bose-Hubbard model with incoherent pumping and losses.

The current manuscript leaves me a bit wondering whether there is something interesting to be said about the first order coherence for this system with respect to the closed system. This leaves one with a nice piece of work where the conclusions seem to be unfortunately a bit trivial. But of course the authors are very much welcome to point out my oversights.

1- Unless I have missed it, the manuscript actually does not point out a big impact of the nonequilibrium condition on the first order correlation function. If the authors cannot convince me that I am wrong, I am afraid that the manuscript lacks a clear message that warrants publication in SciPost.

2- As a side remark, I would like to mention that I did not appreciate very much the discussion of the work in perspective of the NISQ era in the conclusions. There does not seem to be any potential of the presented driven-dissipative Bose-Hubbard model in this respect. It is my opinion that these boiler plate texts should be avoided as much as possible in the ChatGPT era.

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