Quantum circuit refrigerator based on quantum dots coupled to normal-metal and superconducting electrodes

In quantum dot junctions capacitively coupled to a resonator, electron tunneling through the quantum dot can be used to transfer heat between different parts of the system. This includes cooling or heating the electrons in electrodes and absorbing or emitting photons in the resonator mode. Such systems can be driven into a nonequilibrium state by applying either a voltage bias or a temperature gradient across the electrodes coupled to the quantum dot, or by employing an external coherent pump to excite the resonator. In this paper, we present a semiclassical theory to describe the steady state of these structures. We employ a combination of the Floquet-nonequilibrium Green's function method and semiclassical laser theory to analyze a normal metal-quantum dot-superconductor junction coupled to a resonator. Our paper focuses on key parameters such as the average photon number and phase shift in the resonator, the charge current in the quantum dot, and the heat fluxes among different components of the system. We explore how photon-assisted Andreev reflection and quasiparticle tunneling in the quantum dot can refrigerate the resonator mode and the normal-metal electrode. We also examine the influence of finite voltage and thermal biases on these processes.