Current and noise expressions for radio-frequency single-electron transistors

We derive self-consistent expressions of current and noise for single-electron transistors driven by time-dependent perturbations. We take into account effects of the electrical environment, higher-order co-tunneling, and time-dependent perturbations under the two-charged state approximation using the Schwinger-Kedysh approach combined with the generating functional technique. For a given generating functional, we derive exact expressions for tunneling currents and noises and present the forms in terms of transport coefficients. It is also shown that in the adiabatic limit our results encompass previous formulas. In order to reveal effects missing in static cases, we apply the derived results to simulate realized radio-frequency single-electron transistor. It is found that photon-assisted tunneling affects largely the performance of the single-electron transistor by enhancing both responses to gate charges and current noises. On various tunneling resistances and frequencies of microwaves, the dependence of the charge sensitivity is also discussed.