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

TL;DR

This paper models SNAIL-based traveling-wave parametric amplifiers as analogue black-white holes, analyzing their stability and quasi-normal modes to understand their ringdown behavior and nonlinear dispersion effects.

Contribution

It derives the master equation for the probe field, shows the absence of negative modes, and presents the first analysis of quasi-normal modes in this analogue black hole system.

Findings

No normalizable negative modes found, indicating stability.

First semi-analytic and numerical study of quasi-normal modes in this system.

Identifies the timescale for nonlinear dispersion and ringdown excitation.

Abstract

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.