On-demand analog space-time in superconducting networks: grey holes, dynamical instability and exceptional points

1- Compelling motivation for the use of superconducting circuits as analog spacetime geometries. The possibilities seem richer than the actual gravitational metrics.

2- Explicit solutions for the modes in novel 1D array configurations, with Josephson junctions and circulators. This has interest well beyond the application as a simulator.

1- It is not very clear how several of the analogue gravity issues that are thoroughly discussed in the Introduction manifest in the explicit calculations performed later.

2- The proposals require circuit elements, such as gyrators and negative inductances, that are still in early stages of development.

This work proposes the use of novel superconducting circuits as black hole analogs by emulating different spacetime geometries in a controlled manner. Along the way, it provides interesting results, such as calculations for the modes in 1D arrays with gyrators and the development of some technical tools. These result are timely, since the implementation of some of these novel circuit elements is in progress [though this also hinders the applicability of the proposals], and provide a synergetic link with spacetime geometry. However, the manuscript is long and involved, and could benefit from a clarification effort on some issues:

1- The introduction section highlights several potential advantages of the gyrator arrays with respect to other implementations. Could the authors provide a more detailed comparison between platforms [not necessarily in this chapter]? E.g., between the "control over the shape of the space-time geometry" used in convential Josephson junction arrays and the flux quenches. Another related platform is the electromagnetic waveguide of Ref. 46, which just uses (time-dependent) linear elements. Also, in the same work an important issue about the need of quantum effects for Hawking radiation is pointed out [cf. Ref. 9 therein]. Such a discussion, adapted to the proposals of this article, would benefit the comprehensiveness of the introduction.

2- Though not with detail, the authors propose the use of tunable 0-$\pi$ junctions to shift the Josephson potential. However, it shall be noted that typically, in a 0-$\pi$ transition, the ground state changes parity, so the initial and final potentials would not be coherently connected - an external quasiparticle is required to change between such states. Also, could a similar $\phi\sim\pi$ shift be generated by substituting the single junction with an assymmetric SQUID? [even if the effective EJ would change].

3- The authors may specify in which context and in which degree of generality the work "provides a first step towards investigating the backaction of quantum fields on spacetime". The <qualitative speculations> of Sec. 6 seem to indicate that the geometry is dictated/interpreted from a convenient choice that depends on the wavefunctions (e.g. the quadratic expansions of the cosine).

4- Given the importance of the full $\cos \phi$ potential, what is to be expected from the interactions produced by an expansion higher than quadratic?

5- The introductory remarks of Sec. 5 would benefit from a further motivation for the choice of two point correlation function analysis. This could include, e.g., reminding/deepening the reasons given in Sec. 2.2 or a (brief) summary of the methods used in the literature to identify non-trivial geometries and radiation - perhaps moving here part of the last paragraph of Sec. 4.

6- What is the motivation for the choice of the parameters in Figs. 6 and 7? What determines the frequency of the modulation in Fig. 7a? It is a bit surprising that, after such a challenging implementation of the calculations, there is not a more detailed exploration of the system. The comparison between the methods could also be more explicit in Sec. 5 (not only in Sec. 7). Also, while in the bands the EP may pose problems to the direct diagonalization method, in the finite size system it may be that the eigenvalues are not defective. And even if this is the case, why does it prevent to form a complete basis able to describe the dynamics?

The authors may consider the questions posed in the Report, and other minor comments/suggestions in (page number):
(1) "combining" the "nonreciprocity [] with the nonlinearity"
(5) "it is understood since a long time" requires a reference, perhaps a review; "looses"-&gt;"loses" ¿?; can you write the transform back to (t,x) explicitly (for constant u,v)?
(15) Fig. 4. yellow and orange colors are very similar. Vertical lines in $\pm \pi$ could help (also in Fig. 3).
(18) Fig. 5. Label "b)"
(20) "black and white hole horizon"s
(23) "three types", perhaps needs to be more explicit
(27) Eq. A.7. $\mathcal{H}_{1,m}$ ¿?
(30) $2J*2J$ and other instances, perhaps $2J\times 2J$ is more common
(40) Fig. 9a. Labels for $\alpha=0.5, 0.9$ are too similar. Why not draw the theory line for all of them and avoid panel b?
Refs.- Some topics discussed in the article could benefit from a broadened literature research, such as the quantization of nonreciprocal elements or the 0-$\pi$ junctions.

Ask for major revision

The work by Javed et al. proposes an interesting approach to probe some of the physics of black holes in a superconducting circuit. The authors introduce various circuit models comprising some standard circuit elements in cQED and some less standard elements such as gyrators and negative inductances, that add the necessary ingredients to emulate black hole behavior.
I think generally the paper reads nicely and seems scientifically sound. I also find that idea introduced, while possibly rather far from experiments, is intriguing and novel enough to deserve publication.

I have a few questions that I believe could strengthen the paper if addressed.

1) Gyrators introduce a magnetic field in cQED, breaking time-reversal symmetry. It is not clear to me whether one generally needs time-reversal symmetry breaking to simulate black holes. If black holes do not require T-symmetry breaking, while the device that simulates them does, could some aspects of the underlying physics be lost or altered in the simulation?

2) In the device modelled, if the gyrator adds a magnetic field, would this not lead to “Peierls phases” to be accumulated from one unit cell to the next? In this case, I would expect some kind of quantisation condition for the allowed values of G when a periodic system is considered. Could the authors comment on whether such constraints arise in their model, and if so, how they are handled?

3) Since the authors seem to be focusing on a lumped element implementation of the device, it could be helpful to comment on the effect of disorder, especially in terms of variability of the circuit parameters. Could the authors comment on which parameters are most critical to keep uniform for the analogy to hold? A short discussion (or numerical estimate) of the sensitivity of the model to disorder would help guide future efforts.

Publish (meets expectations and criteria for this Journal)