Fluctuations and noise in nanoelectrical and nanomechanical systems

In this thesis we present a study of the fluctuations and noise which occur in a particular nanoelectrical device, the single electron transistor (SET). Electrical transport through the SET occurs through a combination of stochastic, incoherent tunnelling and coherent quantum oscillations, giving rise to a rich variety of transport processes. In the first section of the thesis, we look at the fluctuations in the electrical properties of a SET. We describe the SET as an open quantum system, and use this model to develop Born-Markov master equation descriptions of the dynamics close to three resonant transport processes: the Josephson quasiparticle resonance, the double Josephson quasiparticle resonance and the Cooper-pair resonances. We use these models to examine the noise properties of both the charge on the SET island and the current flowing through the SET. Quantum coherent oscillations of Cooper-pairs in the SET give rise to noise spectra which can be highly asymmetric in frequency. We give an explicit calculation of how an oscillator capacitively coupled to the SET island can be used to infer the quantum noise properties close to the Cooper-pair resonances. To calculate the current noise we develop a new technique, based on classical full counting statistics. We are able to use this technique to calculate the effect of the current fluctuations on an oscillator coupled to the current through the SET, the results of which are in good agreement with recent measurements. In the final part of the thesis we explore the coupled dynamics of a normal state SET capacitively coupled to a resonator in the presence of an external drive. The coupling between the electrical and mechanical degrees of freedom leads to interesting non-linear behaviour in the resonator. We are able to find regions where the resonator has two possible stable amplitudes of oscillation, which can lead to a bistability in the dynamics. We also look at the fluctuations in the energy of the system. We use numerical methods to simulate the dynamics of the system, and to obtain the probability distribution for the work done, whose form can be interpreted by the appropriate fluctuation relation