Reservoir-Engineered Refrigeration of a Superconducting Cavity with Double-Quantum-Dot Spin Qubits

We present an analytically tractable theory of reservoir-engineered refrigeration of a superconducting microwave cavity and map it onto a realistic solid-state implementation based on gate-defined double-quantum-dot (DQD) spin qubits. Treating the DQD not as a spectroscopic element but as a tunable engineered reservoir, we show how gate control of populations, coherences, linewidths, and detuning defines an effective photon birth-death process with predictable detailed balance. This framework yields closed-form expressions for the cavity steady state, identifies cooling bounds and detuning-dependent refrigeration valleys, and clarifies when refrigeration can drive the cavity below both the bath temperature and the DQD setpoint. By distinguishing refreshed (collision-like) and persistent reservoir regimes, we show how memory effects, saturation, and dark-state formation constrain cooling in realistic devices, while collective bright-mode coupling in a two-dot configuration can enhance refrigeration subject to mismatch and dephasing, as confirmed by numerical Lindblad simulations demonstrating targeted millikelvin cavity cooling relevant for cryogenic circuit-QED architectures.