Stability and Quasinormal Modes of Analog Black-White Holes in Superconducting Circuits ✦

Original: "Stability and quasi-normal ringing in analogue black-white holes in SNAIL-based traveling-wave parametric amplifiers"

· Daisuke Yamauchi, Haruna Katayama, Norihiro Tanahashi

arXiv:2605.11565v1 · 2026-05-12 · CC BY 4.0 · ⏱ 2 min · General Relativity Quantum Physics

Perturbations around solitons in superconducting parametric amplifiers decay through a characteristic ringdown — quasinormal modes — confirming the system's stability.

Abstract

The dynamics of a traveling wave in a travelling-wave parametric amplifier (TWPA) based on SNAIL elements is considered, described by the Korteweg–de Vries equation or the modified KdV in the continuous limit. The soliton solution spatially modulates the effective speed of the probe field, realizing an analogue event horizon. A master equation for the weak probe field is obtained, where the background soliton plays the role of a potential. Using methods of supersymmetric quantum mechanics, the absence of normalizable negative modes is shown. For the first time, quasinormal modes of an analogue black-white hole are investigated semi-analytically and numerically; based on the computed frequencies, the timescale for the activation of nonlinear dispersion is established, and the excitation of a ringing mode is demonstrated.

Links in the knowledge graph 1

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.

🎯 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.

-\frac{d^2 H}{d\eta_*^2} + V(\eta_*) H = \epsilon^2 \Omega^2 H

H is the perturbation amplitude, V is the effective potential, η_* is the tortoise coordinate, ε is the stretch parameter, Ω is the quasinormal mode frequency.

V = \frac{1}{\sqrt{v}} \frac{d^2 \sqrt{v}}{d\eta_*^2}

v(η_*) is the spatially modulated velocity of the probe signal; the potential vanishes at the horizon.

\Omega_0 \approx \frac{\beta_{\text{phys}}^{5/4} v_0}{w_{\text{phys}}} (1 - i)

Dependence on the parameter βphys (relative soliton speed), v₀ is the linear velocity, w_phys is the soliton half-width.

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

I am the author