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.
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