Dynamics of a nanomechanical resonator coupled to a superconducting single-electron transistor

TL;DR

This paper analyzes the dynamics of a nanomechanical resonator coupled to a superconducting single-electron transistor near specific resonances, revealing effective thermal behavior, potential instabilities, and analogies to laser cooling.

Contribution

It provides a detailed Fokker-Planck description of the coupled system near resonances and identifies conditions for stability and instability, advancing understanding of nano-electromechanical interactions.

Findings

Resonator behaves as an effective heat bath with a temperature and damping.

Possible mechanical instabilities with negative damping near certain voltages.

Breakdown of the Fokker-Planck approximation close to resonances.

Abstract

We present an analysis of the dynamics of a nanomechanical resonator coupled to a superconducting single electron transistor (SSET) in the vicinity of the Josephson quasiparticle (JQP) and double Josephson quasiparticle (DJQP) resonances. For weak coupling and wide separation of dynamical timescales, we find that for either superconducting resonance the dynamics of the resonator is given by a Fokker-Planck equation, i.e., the SSET behaves effectively as an equilibrium heat bath, characterised by an effective temperature, which also damps the resonator and renormalizes its frequency. Depending on the gate and drain-source voltage bias points with respect to the superconducting resonance, the SSET can also give rise to an instability in the mechanical resonator marked by negative damping and temperature within the appropriate Fokker-Planck equation. Furthermore, sufficiently close to a resonance, we find that the Fokker-Planck description breaks down. We also point out that there is a close analogy between coupling a nanomechanical resonator to a SSET in the vicinity of the JQP resonance and Doppler cooling of atoms by means of lasers.