  We focus on a recently experimentally realized scenario of
normal-metal-insulator-superconductor tunnel junctions coupled to a
superconducting resonator. We develop a first-principles theory to describe the
effect of photon-assisted electron tunneling on the quantum state of the
resonator. Our results are in very good quantitative agreement with the
previous experiments on refrigeration and heating of the resonator using the
photon-assisted tunneling, thus providing a stringent verification of the
developed theory. Importantly, our results provide simple analytical estimates
of the voltage-tunable coupling strength and temperature of the thermal
reservoir formed by the photon-assisted tunneling. Consequently, they are used
to introduce optimization principles for initialization of quantum devices
using such a quantum-circuit refrigerator. Thanks to the first-principles
nature of our approach, extension of the theory to the full spectrum of quantum
electric devices seems plausible.
