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

1- Very comprehensive
2- High quality and clear figures (although see comments on figure 2 below)
3- A decent effort put in to make the paper both precise and accessible.

1- Some loose terminology (see below for comments)

This is a comprehensive piece of work, carefully put together by experts in the field. There is a good balance of specificity (with respect to a particular experiment) and general theoretical development. It is quite long, containing quite a lot of material, and such papers can be challenging to write. I have a number of suggestions/requests for improvements which are essentially all in the nature of improving clarity and tightening up some terminology. However it is a good paper and should be published.

1- Page 1, column 2, paragraph 2: "numerical explore its features" should be "numerically explore its features"

2- Page 2, column 2, equation 1: I think it would be preferable to at least refer to/explain dW as a Wiener noise term a bit sooner after it is introduced.

3- Page 3, column 1, paragraph 2: It is a "Wiener noise term", not "Weiner".

4- Page 4, column 1, paragraph 3: "one or three solutions" could use a bit more precision. A cubic equation always has 3 solutions, however there is a physically motivated criterion here that only non-negative real solutions are meaningful, and it turns out that there is always one and there may be three. I'll also note that I'm a bit surprised that the algebraic solutions aren't just presented, at least in an appendix; they should be fairly straightforward to determine (there's a fairly decent description of the process even on Wikipedia!).

5- Page 4, column 1, equation 19: I think it would be clearer if it was stated that Eq. (18) was differentiated with respect to n_S, and then d J^2/dn_S were set = 0.

6- Page 4, column 2, figure 2: this should be considered optional, however I did note that the various annotations on these plots seemed to be a somewhat inconsistent mixture of times new roman and computer modern (i.e., default LaTeX) fonts. On (d) the ticking on the y axis does not appear even to be the same size as that own the x axis.

7- Page 5, column 2, section IV, paragraph 2: Another optional suggestion. It might be advisable to refer to this method of determining the condensate as an Onsager-Penrose approach; it's not strictly identical to e.g. a symmetry-braking approach of averaging over a field operator.

8- Page 9, column 1, paragraph 1: I am very unsure over the nomenclature "instantaneous chemical potential"! Chemical potential, like temperature, is a strictly equilibrium concept. I think a better way of thinking about what it is that there is a functional of psi which, when psi is a stationary state, gives a value that is a chemical potential. For sufficiently slow timescales, it may be that the concept that a local (in time) "instantaneous" chemical potential is a useful concept, but it's not clear that that is the case here. I would also note that the subscript j appears on the right hand side of the equation, but not the left. Is this an oversight, or is there a summation convention?

9- Page 9, column 2, paragraph 2: Just a comment here regarding "A possible explanation for this outcome is that the coherent condensate atoms tunnel more easily into the system"; this may or may not be a useful analogy, but I cannot help thinking of the superfluid (but not the normal fluid) flowing through a capillary in HeII.

10- page 10, column 1, paragraph 1: I would suggest "stationary states of the GPE" (or something of the sort) as being more appropriate than "eigenmodes".

- Very clearly written
- Thorough analytical and numerical analysis,
- Discussion also of the limitations of the approaches used
- The models provide possible explanations of the experimentally observed effects

- Fine tuning and ad hoc assumptions about the reservoirs are needed to obtain better agreement with experiment

In their paper “Bistability and nonequilibrium condensation in a driven-dissipative Josephson array: a c-field model” Reeves and Davis present a theoretical analysis of an intriguing recent experimental study by the group of Herwig Ott. In this experiment an array of Joesphson junctions is realized using weakly interacting ultracold bosonic atoms in a one-dimensional optical lattice of two-dimensional pancake-shaped traps (sites), each of which hosts a number of relevant transverse states. The system is additionally subjected to controlled local particle loss at the central lattice site. For a wide interval of dissipation strengths, a bistable region is found, where, depending on whether the central dissipative site was initially empty or filled, it remains weakly populated or entirely filled, respectively. While the latter case suggests that a supercurrent into the lossy site is established, the former is interpreted as a regime of incoherent transport. Moreover, it is found that the time the system needs to reach its (quasi)steady state in the normal (non superconducting) branch shows a pronounced maximum as a function of the dissipation strength. This observation was interpreted as possible critical slowing down indicating a phase transition inside the bistable region.

In their paper Reeves and Davis present and solve three different models of increasing complexity to describe the experiment. The first and most simple model, which is introduced to provide a first intuitive picture and reference point, treats the lossy lattice site in a single-mode approximation. It can be solved analytically and shows indeed bistability. Second, a multi-mode model of the lossy site is introduced, using the c-field approach. Here (roughly) only those modes are kept, which, thanks to an occupation larger compared to one, allow for a classical description. This leads to a stochastic Gross-Pitaevskii equation, which is projected onto the (low-energy) subspace spanned by the relevant modes. Here the remaining lattice is captured by a driving term. This approach equally gives rise to bistability. Moreover, it provides also a possible explanation of the transition (or crossover) and critical slowing down happening in the bistable region. Namely, it is associated with the formation of a large occupation of several excited modes which are in resonance with the reservoir (which the authors call quasicondensate). While offering an explanation for various experimentally observed features, the single-site c-field model fails to provide quantitative agreement. Therefore, it is extended by making the reservoir dynamical. Both the left and the right reservoir are treated using a phenomenological multi-mode model coupled to an effective thermal reservoir. By tuning the effective temperature of the bath, quantitative agreement with the experiment can be established for this third model. However, this model does not give rise anymore of the tradition (“quasicondensation”) within the bistable region.

The paper provides a thorough analysis of the non-equilibrium behavior of a dissipative quantum system. It is very well written. The authors provide the right amount of details (not too few not too many) and the way the paper is structured from simple to more elaborate models works well. Also the limitations of the methods used are discussed. The discussion of the models and their properties (obtained analytically and using the c-field approach) is well organized and clear. Remaining discrepancies between theory and experiment as well as ad hoc assumptions regarding the design of the dynamical reservoirs and the need to fine tune the effective temperature to achieve better agreement suggest, however, that the models do not yet fully explain the physics of the experimental system. But his is fine. I am convinced that the paper provides interesting insights into the physics of the system and possible mechanisms explaining its features. Before recommending publication, I would, however, kindly ask the authors to address the following (mostly minor) points.

1) Is it obvious to base the truncation on the enemies of the single-particle orbitals? For Bose condensates in a trap it is well known that the Thomas-Fermi radius can differ substantially from the harmonic oscillator ground state. Does the Thomas Fermi-Radius provide a rough estimate for the number of states needed to be taken into account?

2) I do not link the use of the term “quasicondensate” very much. This word has been introduced for specific superfluid equilibrium states, featuring quasi-long range order (i.e. algebraic decay of the single-particle coherences). Maybe there is a better way of calling this state.

3) I find the non-equilibrium state which is called quasicondensate very interesting. I think it would be desirable, if the authors could provide more details about it and how the transition/crossover to the normal phase occurs.

4) Please specify the gamma values used in the right panels of Fig. 3. (Here also a series of such plots showing the transition from normal via “quasicondensate” to condensate at more intermediate points would be very interesting, see 3).

5) Maybe g^(2) should be called two-particle correlations (rather than coherences).

6) In Fig. 5.(c) the lower phase boundary the label SF/ NS seems to be wrong.

7) I think the choice of the effective temperature should be explained in more detail. Is there any rational behind choosing a particular values, besides tuning the phase boundary to the experimental value?

8) Maybe, I overlooked it, but I did not find information about how the yellow star symbol in Fig. 10 were obtained. Probably they give information about the metastable “quasicondensation”. I think it would be very nice to see some data indicating this transient phenomenon.