Classical dynamics of a nanomechanical resonator coupled to a single-electron transistor

Abstract

We analyze the dynamics of a nanomechanical resonator coupled to a single-electron transistor ͑SET͒ in the regime where the resonator behaves classically. A master equation is derived describing the dynamics of the coupled system which is then used to obtain equations of motion for the average charge state of the SET and the average position of the resonator. We show that the action of the SET on the resonator is very similar to that of a thermal bath, as it leads to a steady-state probability distribution for the resonator which can be described by mean values of the resonator position, a renormalized frequency, an effective temperature, and an intrinsic damping constant. Including the effects of extrinsic damping and finite temperature, we find that there remain experimentally accessible regimes where the intrinsic damping of the resonator still dominates its behavior. We also obtain the average current through the SET as a function of the coupling to the resonator.

Self-consistent theory of molecular switching

“…This last term corresponds to the dissipation induced by the retardation of the electronic degrees of freedom, which do not respond immediately to a change in x ͑first nonadiabatic correction͒. 25 It can also be traced to the "quantum" nature of the charge noise, i.e., a slight asymmetry between the charge noise at positive and negative frequencies. [32][33][34] As a result, the dynamics of the mode x becomes essentially classical, described by the Langevin equation, 14…”

“…11 In this paper, we study the case of "slow" phonons at strong coupling, ⌫ӷ 0 for eV Ͼប 0 . 12,14,[25][26][27] The physical distinction between this case and the one of "fast" phonons, ⌫Ӷ 0 , can be understood in the following way: For fast phonons, every electron tunneling event occurs over many oscillator periods. Thus effectively electrons can only couple to ͑or "measure"͒ the energy ͑i.e., occupation number͒ of the oscillator.…”

Noise in a superconducting single-electron transistor resonator driven by an external field

“…Inserting Eq. ͑15͒ into this gives an expression for the charge consisting of a systematic part, and a fluctuating part containing the noise operators 02 , 11 , and 22 . We now go on to calculate the fluctuation spectrum of the charge due to the systematic and noisy terms.…”

“…Such back-action dynamics have also received considerable attention in the context of a mesoscopic conductor used to investigate the behavior of a mechanical resonator. [6][7][8][9][10][11][12][13] One system that is of particular interest is that of a superconducting single-electron transistor ͑SSET͒ coupled to a resonator, either mechanical [14][15][16][17][18][19][20] or composed of a superconducting stripline. 21 The coherent transport through this device at the Josephson quasiparticle resonance 22,23 ͑JQP͒ allows a very low-noise current and, at the same time, the sensitivity of the SSET to charge means that the resonator-SSET coupling is significant.…”

Suppression of the stochastic fluctuations of suspended nanowires by temperature-induced single-electron tunneling

“…In spite of the difficulties related to the definition of temperature for systems that are out of thermodynamic equilibrium, several theoretical works show that nonequilibrium fluctuations in the properties of nanomechanical oscillators coupled to mesoscopic electronic systems (such as a single-electron transistor or a superconducting Cooper pair box) are, to a good approximation, still described by Gaussian distribution functions [4,5].…”