Figure 3. Quantum-circuit refrigeration and thermal model.

(a) Experimentally measured changes in the electron temperatures of the QCR, ΔT_QCR (purple circles) and of the probe resistor, ΔT_probe (red circles), as functions of the refrigerator operation voltage V_QCR. The dashed lines show the theoretical ΔT_probe with (black) and without (green) photon-assisted tunnelling. (b) Tunnelling diagram similar to that in Fig. 2a, but for a higher operation voltage corresponding to the region highlighted in yellow in a. Here only photon emission to the resonator is suppressed by the lack of thermal excitations. (c) Thermal model used for the experiment. Blue colour denotes the electron system of the QCR and red colour that of the probe resistor. Only the fundamental mode of the resonator is considered. The power P_T arises from photon-assisted tunnelling; and correspond to ohmic losses; P_ep accounts for coupling between the probe electrons and the phonon bath at temperature T₀; P_res denotes the residual heating power of the probe due to V_QCR; P_leak accounts for leakage of photons to the resonator from high-temperature stages of the cryostat; and P_x denotes excess power due to a constant thermal conductance G_x to a reservoir at temperature T_x. Negative power implies the opposite direction of the energy flow with respect to the shown arrows. See Supplementary Note 2 for a detailed description of the model. (d) Resonator temperature (T_res, grey line) and average photon number (n, red line) solved from a thermal model corresponding to the measurements in a. See Supplementary Note 6 and Supplementary Fig. 4 for more information and for data at different bath temperatures.