Quantum master equation descriptions of a nanomechanical resonator coupled to a single-electron transistor

TL;DR

This paper derives and analyzes quantum master equations for a nanomechanical resonator coupled to a single-electron transistor, revealing how the resonator's damping and frequency shift depend on system time-scales and providing simplified models for its dynamics.

Contribution

The paper introduces a microscopic derivation of master equations for the coupled system and compares different reduced models, enhancing understanding of resonator behavior in quantum regimes.

Findings

Resonator behaves as a damped harmonic oscillator with a shifted frequency.

Maximum damping occurs when mechanical and electrical time-scales match.

Frequency shift is largest when the resonator moves much slower than the charge dynamics.

Abstract

We analyse the quantum dynamics of a nanomechanical resonator coupled to a normal-state single-electron transistor (SET). Starting from a microscopic description of the system, we derive a master equation for the SET island charge and resonator which is valid in the limit of weak electro-mechanical coupling. Using this master equation we show that, apart from brief transients, the resonator always behaves like a damped harmonic oscillator with a shifted frequency and relaxes into a thermal-like steady state. Although the behaviour remains qualitatively the same, we find that the magnitude of the resonator damping rate and frequency shift depend very sensitively on the relative magnitudes of the resonator period and the electron tunnelling time. Maximum damping occurs when the electrical and mechanical time-scales are the same, but the frequency shift is greatest when the resonator moves much more slowly than the island charge. We then derive reduced master equations which describe just the resonator dynamics. By making slightly different approximations, we obtain two different reduced master equations for the resonator. Apart from minor differences, the two reduced master equations give rise to a consistent picture of the resonator dynamics which matches that obtained from the master equation including the SET island charge.