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

We thank the referee for the positive assessment and careful reading of the manuscript. We believe that in responding to the referee’s various very justified questions and suggestions, our paper has been significantly improved.

The answers to referee's questions can be found below, for an overview of the changes made to the paper please see the "List of changes" to be submitted with the revised manuscript.

1) In other proposals for simulating spacetime geometry in circuit QED [32,33,46], to recreate a spacetime with anisotropy in signal speeds but no horizon, already requires a time dependent electromagnetic field that sweeps through the circuit. This leads to the issue that even a stationary spacetime (i.e. spacetime with time translation symmetry), such as a spacetime with no apparent horizons, requires a time dependent circuit which is not the case when using gyrators. Additionally, when investigating a non-stationary spacetime, such as one with an evaporating horizon where we do use a time-dependent flux, using gyrators has the advantage of fixing the position of the horizons in the lab frame as opposed to a co-moving frame that moves with the sweeping electromagnetic field. We have added a remark to this effect in Section 2.2, on Page 8.

2) We thank the referee for pointing out this caveat about fermion parity. We do not expect this aspect to be critical for our proposal, as different fermion parity sectors can be incorporated in circuit QED through periodic and anti-periodic boundary conditions in ϕ space. Additionally, the boundary conditions do not play a role in the here considered regime of large Josephson energies. Furthermore, our proposal does not necessitate coherent superpositions between the 0 and \Pi configurations, such that we do not expect any conflict with the Wigner superselection rule. The main bottleneck requirement is a sufficiently fast swap from 0 to \Pi configuration, such that the many-body wavefunction is located at the maximum of the cosine immediately after the quench.

Finally, the referee is correct in expecting that an asymmetric SQUID works as well in implementing a \Pi shift. It has to be noted though that the mechanism is exactly the same as for a single Josephson junction, and moreover in a generic SQUID device the efficiency of the conversion from the current amplitude to the phase shift is reduced. This can be easily seen when considering the SQUID Josephson energy as a sum of two shifted cosines, where the external current couples with a likewise asymmetric coupling prefactor to both the strong and the weak junction. The effective shift of the total Josephson energy depends on the asymmetry of this coupling. The efficiency is maximal if the current couples exclusively to the larger of the two junctions and minimal when it couples to the smaller junction. We have added a small calculation (paragraph surrounding Eq. (20)) showing this on Page 20.

3) The answer to this question is strongly related to our answer to referee’s next question. Please consider our answer below to point 4.

4) The full cos(ϕ) potential plays a special role in a limited sense, we need it in order to induce a \Pi shift in the potential and create negative inductances. However we consider a regime of large Josephson energies w.r.t. other energy scales, i.e. before and immediately after the quench the many-body wavefunction remains localized justifying the quadratic expansion of the cosine. The non-linearity of the cosine will without a doubt play a crucial role beyond transient times, as also discussed in Sec. 6 of the manuscript. We sincerely hope the referee understands that this is an extremely complex question that requires an entirely dedicated new research effort. For instance non-linear corrections will give rise to interacting bosons and potentially quantum phase-slip physics. We point to that possibilities in an additional phrase in the outlook.

Based on this we can also answer referee’s question in point 3 above. The referee correctly understood that our approach relies on taking a reference point in the Hilbert space spanned by the superconducting phases and expanding around it. In the wormhole regime, this expansion is by definition unstable, due to exponential accumulation of quantum fluctuations, which cause problems beyond transient times, due to non-linearity. As we pointed out above in our response, the full solution to this interacting (non-linear) problem is hard to obtain. But including dissipation, which can qualitatively be captured by fuzzy projections in the space of superconducting phases, the problem can be simplified again, expanding around different minima (after environment-induced projection), leading to a (time-dependent) diffusion of spacetime, and thus inevitably to backaction, since the expanded (nearly linear) scalar field always reacts to the updated spacetime configuration. In fact, in this regard, our expansion approach might actually not only serve as a simulation platform, but conversely, the diffusive spacetime perspective could also serve as a useful conceptual trick to capture strongly non-linear systems in the presence of dissipation. We have added a remark to this end in the outlook.

With the unique use of non-linearity specific to our work, and with the potential connection to backaction on space-time geometry, we believe that our work highlights an important, and previously disregarded regime of non-linear devices prepared in a precarious unstable starting state, suitable for multi-pronged future research efforts.

5) We thank the referee for this remark to improve the motivation for using two-point correlation functions. We chose to follow their second suggestion and refer to previous literature using two-point correlation functions to examine features related to analog event horizons. The beginning of Section 5 now contains an additional text passage referring specifically to experiments done in cold atoms (predominantly Steinhauer’s work), where two-point correlation functions were central to probe the thermal nature of analog Hawking radiation.

6) We have added a more detailed discussion of dispersion relation in Eq. (20) on Page 14, pointing out that there are only two dimensionless free parameters, the gyration conductance G and the ratio of the inductive energy to the charging energy E_L/E_C. While the latter has to be larger than 1 to justify the harmonic approximation of the cosine, there is per se no restriction on G (up to a quantization constraint due to phase compactness, see also other referee report). However, the relative magnitude of E_L/E_C and G is not of importance for the creation of analog event horizons – the only thing that matters is the sign of E_L/E_C. The choice of parameters in Figs. 6 and 7 merely reflects this observation.

The frequency of modulation in Fig. 7a is a consequence of the quench and is independent of the presence of the horizons, hence this feature is still present when the quench is performed in the entire circuit (i.e. no horizons). We expand on this in added text on Page 23 and in the newly added Fig. 8.

The existence of EPs in the parameter space of the system, boils down to checking if the relevant matrix is diagonalizable or not, for every value of system parameters. The check if a given matrix is diagonalizable, apart from special cases where the spectral theorem is applicable, is in general numerically expensive (for large matrices) and not very accurate if the eigenvalues are very close to each other. Hence, it is better to bypass this procedure and not worry about diagonalizing a possibly defective (non-diagonalizable) matrix. Additionally, it follows from the definition of diagonalizability that a defective matrix cannot have a complete basis of eigenvectors. One can, in principle, work with generalized eigenvectors, but this again brings us back to the problem of first determining whether a given matrix is defective or not. We have added additional text in Section 5 explaining the motivation for using Klich’s determinant formula.

We warmly thank the referee for the favorable report and finding our ideas intriguing and novel. We believe that our revision of the manuscript in response to the referee’s comments have allowed us to increase the depth of our work.

The answer to the referee's questions are formulated below, for a detailed overview of the changes made to the paper please see the "List of changes" to be submitted with the revised manuscript.

1) Considering simulations of apparent horizons as originally proposed by Unruh (see also the review by Robertson, Ref. (43)) breaking time-reversal symmetry (TRS) is necessary. Typically, the goal is to overtilt the dispersion relation making both branches travel in the same direction. Any non-zero tilt (i.e. non-zero v in Eq. (2)) invokes terms that break time-reversal symmetry, specifically the terms linear in time derivatives of the scalar field. While other propositions within circuit QED (mostly the works dealing with solitons Ref. (33)) do not seem to require explicit TRS breaking elements, the emergent effective Lagrangian still violates TRS as it is formulated in the co-moving soliton frame. Our work therefore differs from previous works by incorporating TRS breaking via an explicit circuit element. This gives us a better control of the effective spacetime geometry, allowing us to study various unconventional features such as: grey holes and their connection to exceptional points. In the manuscript we have added a corresponding footnote (Footnote 1).

2) We thank the referee for this intriguing question. The referee is correct with their expectation that G takes on discrete values, when considering compact (2 periodic) superconducting phase (ϕ) values for each node. This is simply the torus version of Dirac’s well-known magnetic monopole quantization argument. We are acutely aware of this feature, and some of us have already embarked on an extensive research endeavor exploring a number of fascinating consequences due to quantized gyration conductance (see Ref. (72)). However, in order for the quantization of G and the compactness of ϕ to become relevant, the system at low energies needs to be able to explore the full ϕ space. We consider a regime where the Josephson energies are high enough such that phase differences remain in the troughs of the cosine potential. In that case the value of G does not necessarily correspond to the Chern number defined as the integral of the Berry curvature over the entire (2 periodic) ϕ space, but instead corresponds to local expansion of the Berry curvature close to the energy minima. This local Berry curvature does not need to yield quantized values, and in addition can continuously fluctuate from one gyrator element to the next. This motivated us to include disorder in the gyration value, also in response to the referee’s third question. In case the referee is interested in learning about the detailed consequences in the opposite regime of weak Josephson effects, we recommend considering our previously mentioned work on the topic, where quantized G (even inhomogeneous G values) are considered and predicted to give rise to anyonic excitations with strange non-local features. We have added a footnote (Footnote 3) to this effect.

3) We have added the Appendix F and additional text at the end of Sec. 5, briefly discussing the effect of disorder in the gyrator and inductive elements. The main result of this investigation is that for relative disorder strength of 10% and below, the evaporation of horizons in both the circuits is insignificantly affected. For the same disorder the quantum fluctuations in the wormhole region remain unchanged for the circuit with only nearest neighbor connections but show some deviation from the case of no disorder for the circuit where JJs connect next nearest neighbors. We also find that for similar relative strength of disorder, the disorder in gyrator elements is more impactful than the disorder in inductive elements. This can be understood by looking at Eq. (20) in the manuscript combined with the fact that the parameter regime we work in requires 4G^{2}>E_{L}/E_{C}, this leads to any change in gyrator conductance being more effective than a change in inductance.