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

The circuit dynamics constructed by traveling-wave parametric amplifiers (TWPA), using superconducting nonlinear asymmetric inductive 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, in the effective theory where the nonlinear dispersion is neglected, 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 in the dispersionless effective theory by numerical and semi-analytic methods. Based on the resultant QNM frequency, we clarify the timescale at which nonlinear dispersion becomes effective in the SNAIL-TWPA circuit system and suggest the possibility of observing ringdown.