Stability and Quasinormal Modes of Analog Black-White Holes in Superconducting Circuits ✛ ✦
Context
Analog black holes, realized in superconducting circuits, offer a unique opportunity to test strong gravity regimes without astrophysical observations. Previously, Hawking radiation has been simulated in Bose-Einstein condensates and optical fibers. The new work focuses on black-white holes in traveling-wave parametric amplifiers (TWPAs), where solitons create an effective curved spacetime. For precise predictions, understanding the evolution of small perturbations on such a background is crucial for future laboratory quantum gravity experiments.
Methods
The foundation is the derivation of a Schrödinger-like equation for the probe field. A transformation to coordinates similar to the tortoise coordinates of Schwarzschild and elimination of the first derivative yields an effective volcano-shaped potential. Stability is proven by the S-deformation method using supersymmetric generators — an idea tracing back to Penrose's work on global horizon structure. The quasinormal mode (QNM) frequencies are found semi-analytically (sixth-order WKB approximation with Padé approximants) and by direct numerical shooting validated against the Pöschl–Teller potential (relative error 10⁻⁷).
Results
It is shown that there are no exponentially growing modes — the system is stable. The least damped QNM is purely imaginary: for example, for a KdV soliton with speed parameter βphys=0.3, the damping frequency |Im Ω| ≈ 0.70 v₀/$w_{phys}$, where v₀ is the linear speed and $w_{phys}$ the physical half-width of the soliton. This means the ringdown lasts several cycles before nonlinear dispersion becomes noticeable. Estimates of nonlinear corrections show they are suppressed by a factor βphys far from the horizon, though they grow near the redshifted horizon. Higher-order QNMs also lie on the imaginary axis, unlike classical black holes where frequencies have a real part.
Implications
The results lay the groundwork for analog gravity experiments: the predicted QNMs can be detected in real SNAIL-TWPA chips. This also paves the way for creating stable 'black hole lasers' and testing quantum effects in curved spacetime.
Future development
Experimental verification of QNMs is expected: observing the damped ringdown and measuring its duration. Theoretically, it is important to study the influence of quantum fluctuations and the precise role of nonlinear dispersion near the horizon, and to extend the analysis to moving solitons.
Impact
The work will impact quantum simulations of gravity, nonlinear dynamics in superconducting metamaterials, and the development of parametric amplifiers with controlled dispersion.
Next steps
Next steps include developing more accurate numerical methods (e.g., Leaver's method) for small βphys, and designing experiments to excite and measure QNMs in SNAIL systems.
Key open problems
The study is directly linked to the unsolved problem of quantum gravity — via laboratory simulation of the Hawking effect — and also to the question of wormhole stability, for which the black-white hole serves as a one-dimensional analogue.
The knowledge machine's view
This section is addressed to the author of the paper and their colleagues. Neighbouring works were found by meaning-based search across our archive, not by word overlap. We do not review: we show the field around the work.
What the work does
This work presents a model of an analog black-white hole in a superconducting parametric amplifier (TWPA) based on SNAIL elements, where a soliton solution of the KdV/mKdV equations creates an effective horizon. The stability of the background is investigated, and the quasinormal modes of a probe field are computed using S-deformation, WKB, and a numerical shooting method. No growing modes are found, and the least damped mode turns out to be purely imaginary.
Where its strength is
The combination of supersymmetric quantum mechanics analytical methods with high-precision numerical calculations provides reliable proof of stability and accurate values of quasinormal frequencies, laying a solid foundation for experimental verification in real chips.
Where it could go next
- One could compare the quasinormal mode spectrum of the analog black-white hole with the behavior of overtones in gravitational waves, where nonlinear contributions have recently been discovered.based on: Gravitational Waves Gain Overtones arXiv:2510.16903
- It is interesting to consider how quantum corrections to the effective potential might alter mode damping, analogous to how quantum effects modify the ringdown of astrophysical black holes.based on: How a Quantum Twist Changes a Black Hole's Ring arXiv:2607.22967
- A calculation of the system's response to an external perturbation is possible, which would link theoretical modes to a real measurable signal in the TWPA.based on: A Black Hole Listens: How Sound Emerges from Disturbance arXiv:2605.24704
Worth trying
- Check whether nonlinear overtones appear in the analog system when the probe field amplitude is increased.
- Assess the impact of quantum fluctuations on the decay time of the purely imaginary mode.
- Simulate an experiment exciting the QNMs with a short pulse and compare with the predicted ringdown.
Why this may matter
If the predicted quasinormal modes are detected in superconducting circuits, this would be the first laboratory confirmation of an analog horizon's ringdown and would open the way to testing quantum effects in strong gravity.
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Original abstract
as written by the authors arXiv:2605.11565v1
The circuit dynamics constructed by traveling-wave parametric amplifiers (TWPA), using superconducting nonlinear asymmetric elements (SNAILs), are known to be approximately described by the Korteweg-de Vries (KdV) or modified KdV equations in the continuum limit and admit soliton solutions. The soliton spatially modulates the effective propagation velocity of the weak probe field, which leads to the effective realization of the causal structure of the analogue event horizons in the SNAIL-TWPA circuit system. In this paper, we derive the master equation for the weak probe field where the background soliton acts as an effective potential. We show the absence of normalizable negative modes in the SNAIL-TWPA circuit system by using the language of supersymmetric quantum mechanics. We also present the first study of quasi-normal modes (QNM) of the SNAIL-TWPA analogue black-white hole system by semi-analytic and numerical methods. Based on the resultant QNM frequency, we clarify the timescale at which nonlinear dispersion becomes effective in the SNAIL-TWPA circuit system and demonstrate how ringdown is excited.
SNAIL stands for 'Superconducting Nonlinear Asymmetric Inductive eLement,' but in English snail is, well, a snail. So fast microwaves here slow down to a snail's pace in tortoise coordinates!
Analog black-white holes resemble wormholes from science fiction (like in the movie Interstellar), connecting distant regions of spacetime.
Key numbers
- Damping frequency of the fundamental mode (KdV, β=0.3): 0.70 v₀/w_phys
- Number of observable cycles before nonlinear breakdown: 3–5
- Typical soliton speed parameter βphys: 0.3
- Numerical method accuracy (Pöschl–Teller test): 10⁻⁷
- Order of WKB corrections for βphys>0.3: <1