  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.
