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

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 timescales 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.

Lasing in circuit quantum electrodynamics with strong noise

“…1(a)]. This system has been theoretically studied [3][4][5][6][7] and it has been experimentally shown [1,2] that lasing can be achieved. Theoretically it has also been shown that an SSET can be used to create a nonclassical photon distribution in a resonator [8,9].…”

Semiclassical dynamics of nanoelectromechanical systems

“…This has turned out to be a highly successful approach, in particular to describe such various systems as NEMS coupled to single electron transistors (SETS) [9][10][11] , FranckCondon blockades 12 in transport through molecules with strong electron-phonon coupling, or quantum shuttles [13][14][15][16] . From the theory of electronic transport through nanostructures 17 , however, it is known that such approximations usually are reliable only in the limit of high external voltage bias, where non-Markovian effects 18 due to quantum coherences between the external reservoirs and the electronic system (SET, quantum dot etc.)…”

Phonon distributions of a single-bath mode coupled to a quantum dot

“…Such an approach has already been employed in the past for describing systems such as a metallic [22] or a superconducting single electron transistor [23].…”

“…In the following, terms stemming from the principal value are neglected since they lead to small corrections in the perturbative regime [22,40]. Denoting withρ n I,qq (t) = n, q|ρ I (t)|n, q , the GME has the compact form (here and in the following, 0 n 2 unless stated otherwise)ρ…”