Time-dependent currents of a single-electron transistor in dissipative environments

Currents of the single-electron transistors driven by time-dependent fields via external dissipative circuits are investigated theoretically. By expressing the external circuit in terms of driven harmonic oscillators and using the reduced-density operator method, we derive time- and environment-dependent tunneling rates in the regime of sequential tunneling and present expressions for both displacement and tunneling currents with these tunneling rates. It is found that the dissipative environments affect tunneling currents in two ways; the determination of driving voltages at tunneling junctions and the depletion of particle-hole distribution functions. Considering a simple dissipative circuit, we discuss the effects of the environment on tunneling currents in both static and time-dependent cases.