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

TL;DR

The paper develops a programmable analog-spacetime platform using superconducting circuits built from Josephson junctions, gyrators, and inductors to realize on-demand lattice metrics and horizons. By engineering a local tilt in the dispersion via nonreciprocal gyrators and transient flux-driven negative inductance, the authors demonstrate horizon formation with either grey-hole or pure black/white-hole character and reveal intrinsic lattice instabilities tied to EPs, challenging conventional Hawking-radiation expectations. They provide two circuit models (nearest- and next-nearest-neighbor couplings), analyze dynamical signatures through two-point correlations and Bogoliubov/Klich methods, and discuss the long-time fate including wormhole evaporation and environment-driven dissipation. The work offers a highly tunable, experimentally accessible route to explore analog gravity, EP physics, and backaction between quantum fields and emergent spacetime geometry, with concrete pathways to small-scale proofs-of-principle.

Abstract

There has been considerable effort to mimic analog black holes and wormholes in solid state systems. Lattice realizations in particular present specific challenges. One of those is that event horizons in general have both white and black hole (grey hole) character, a feature guaranteed by the Nielsen-Ninomiya theorem. We here explore and extend the capability of superconducting circuit hardware to implement on-demand spacetime geometries on lattices, combining nonreciprocity of gyrators with the nonlinearity of Josephson junctions. We demonstrate the possibility of the metric sharply changing within a single lattice point, thus entering a regime where the modulation of system parameters is "trans-Planckian", and the Hawking temperature ill-defined. Instead of regular Hawking radiation, we find an instability in the form of an exponential burst of charge and phase quantum fluctuations over short time scales - a robust signature even in the presence of an environment. Moreover, we present a loop-hole for the typical black/white hole ambiguity in lattice simulations: exceptional points in the dispersion relation allow for the creation of pure black (or white) hole horizons, at the expense of a radical change in the dynamics of the wormhole interior.

Figures (15)

• Figure 1: a) Schematic of a gyrator, a four-terminal circuit element, introduced by Tellegen tellegen_1948. Here two of its terminals have been grounded, hence it acts as an effective two terminal element. b) One possible realization of a (quantum) gyrator, as proposed in kashuba2024gyratorsanyons, by using quantum dots proximitized with superconductors. Building up on the prediction of Ref. Riwar_2016 that conventional multiterminal Josephson junctions harbour topological bands in the space of superconducting phases, one of the goals of Ref. kashuba2024gyratorsanyons was the development of blueprints to create topological flat bands, where the circuit behaviour reduces to Eqs. \ref{['eq_transport_gyrator1']} and \ref{['eq_transport_gyrator2']}. The above quantum dot array is one of three proposed ways to generate topological flat bands in multiterminal junctions. Flat bands emerge under appropriate tuning of the central flux ($\phi_0$) and for sufficiently long chains ($J$ large). For further details, see Ref. kashuba2024gyratorsanyons.
• Figure 2: Wormhole simulations with quantum circuits. In circuits a) and b) (nearest and next-to-nearest Josephson coupling, respectively) the presence of gyrators tilts the dispersion relation giving left and right moving wave packets different speeds. The circuit c) is obtained by inverting the signs of a finite connected region inductances of circuit a). This is achieved by a flux quench induced by a localized current (inset), shifting the superconducting phase difference across the junction by $\pi$, leading to negative inductance. This creates two boundaries between the normal and over-tilted regions, which act as the apparent horizons. Horizons for the circuit with Josephson junctions connecting next-nearest neighbors b) can be obtained in a similar manner.
• Figure 3: The above two figures show that the low-momentum eigenvalues (blue lines) of the circuit Hamiltonians \ref{['eq:model_ham_1']} and \ref{['eq:model_ham_2']} can be approximated by the linear dispersion relation \ref{['eq:dispersion_relation_linear']} (orange lines) at low energies. The branches of the dispersion relation correspond to right and left movers with speeds $u\pm v$. One crucial point about the eigenvalues in b) is that near $k=\pi$ the dispersion relation (brown lines) is the mirror image of the dispersion near $k=0$, which is a feature that the eigenvalues in a) do not have. Parameters used: $E_\text{C}/E_{L} = 1.3, E'_{L}/E_{L} = 0.25,G = 0.6$, and $J = 50$. To lift the degeneracy at zero energy, a small mass term is used $(\approx 10^{-3}E_{L}$).
• Figure 4: The principle behind the topological loop-hole for black versus white hole horizons in lattice systems via EPs. In panels a) and b), we show the complete dispersion of the circuit Hamiltonians \ref{['eq:model_ham_1']} and \ref{['eq:model_ham_2']} with negative inductances. When the Josephson junctions connect nearest neighbors, a), the eigenspectrum is complex (blue line represents real part of the eigenvalues and the green line the imaginary part). The points where the eigenvalues go from real to complex are EPs (marked by red dots). Consequently, the spectrum only crosses zero once at $k=0$, and satisfies the periodicity constraint of the Brillouin zone via a detour in the complex plane. For next nearest neighbor coupling, b), the spectrum is real for all values of $k$, and crosses zero twice, again with mirror images at $k=0$ and $k=\pi$. The low momentum eigenvalues (orange line) for both the circuits do show an overtilt, which is required to create a horizon. Parameters are same as in Fig. \ref{['fig:dispersion_relations']}, except the signs for $E_{L}$ and $E_{L}'$ are flipped.
• Figure 5: Illustration of non-local effects of the current loop flux drive. In order to create the target phase shift profile of exactly $\pi$ within a connected region in the junction array (from $j=j_0$ to $j=j_1$), the corresponding applied current profile as a function of site index $j$ needs to be slightly nonlocal. We have plotted three current profiles, that were obtained for a system with 50 sites with $j_{0}=15$ and $j_{1}=35$, with different ratios of $\Delta l$ to $\Delta x$ as shown in the graph. The parameter $R$ does not have a significant effect on the nature of the current profile, only on the magnitude of the ration $I_{j}/I_{0}$, hence we set $R/\Delta x=1$.