  Previous studies of photon-assisted tunneling through
normal-metal-insulator-superconductor junctions have exhibited potential for
providing a convenient tool to control the dissipation of quantum-electric
circuits in-situ. However, the current literature on such a quantum-circuit
refrigerator (QCR) does not present a detailed description for the charge
dynamics of the tunneling processes or the phase coherence of the open quantum
system. Here we derive a master equation describing both quantum-electric and
charge degrees of freedom, and discover that typical experimental parameters of
low temperature and yet lower charging energy yield a separation of time scales
for the charge and quantum dynamics. Consequently, the minor effect of the
different charge states can be taken into account by averaging over the charge
distribution. We also consider applying an ac voltage to the tunnel junction,
which enables control of the decay rate of a superconducting qubit over four
orders of magnitude by changing the drive amplitude; we find an
order-of-magnitude drop in the qubit excitation in 40 ns and a residual reset
infidelity below $10^{-4}$. Furthermore, for the normal island we consider the
case of charging energy and single-particle level spacing large compared to the
superconducting gap, i.e., a quantum dot. Although the decay rates arising from
such a dot QCR appear low for use in qubit reset, the device can provide
effective negative damping (gain) to the coupled microwave resonator. The Fano
factor of such a millikelvin microwave source may be smaller than unity, with
the latter value being reached close to the maximum attainable power.
