Dynamical instabilities of a resonator driven by a superconducting single-electron transistor

TL;DR

This paper studies the dynamical instabilities of a resonator coupled to a superconducting single-electron transistor at the Josephson quasiparticle resonance, using a semiclassical approach to predict limit cycles and transitions.

Contribution

It introduces a mean field framework to analyze resonator instabilities coupled to an SSET, aligning well with full quantum solutions for weak coupling.

Findings

Limit cycle states depend on coupling, detuning, and frequency.

Mean field equations predict the location and size of instabilities.

Qualitative agreement with full quantum solutions for weak coupling.

Abstract

We investigate the dynamical instabilities of a resonator coupled to a superconducting single-electron transistor (SSET) tuned to the Josephson quasiparticle (JQP) resonance. Starting from the quantum master equation of the system, we use a standard semiclassical approximation to derive a closed set of mean field equations which describe the average dynamics of the resonator and SSET charge. Using amplitude and phase coordinates for the resonator and assuming that the amplitude changes much more slowly than the phase, we explore the instabilities which arise in the resonator dynamics as a function of coupling to the SSET, detuning from the JQP resonance and the resonator frequency. We find that the locations (in parameter space) and sizes of the limit cycle states predicted by the mean field equations agree well with numerical solutions of the full master equation for sufficiently weak SSET-resonator coupling. The mean field equations also give a good qualitative description of the set of dynamical transitions in the resonator state that occur as the coupling is progressively increased.