cycle that leads to population inversion, we show that one can pump population from the qubit excited state $|0L \rangle$ to the qubit ground state $|0R\rangle$ (Fig. \[fig:4\]A) via an ancillary energy level $|1R\rangle$. In the case where the population in $|0L \rangle$ results from thermal excitation, the transfer of population to $|0R\rangle$ effectively cools the qubit by lowering its effective temperature. This kind of active cooling represents a means to initialize and reset qubits with high fidelity, key elements for quantum information science and technology. Alternatively, the pumping mechanism can be used to refrigerate environmental degrees of freedom [@Niskanen07b], or to cool neighboring quantum systems [@You08a; @Grajcar08a].

More explicitly, for a qubit in equilibrium with its environment, the population in $|0L\rangle$ that is thermally excited from $|0R\rangle$ follows the Boltzmann relation $$p_{0L}/p_{0R} = \exp[- \varepsilon /k_{\rm B} T_{\rm bath}],
 \label{Boltzmann}$$ where $p_{0L,0R}$ are the qubit populations for energy levels $\varepsilon_{0L,0R}$, $\varepsilon = \varepsilon_{0L}- \varepsilon_{0R}$, $k_{\rm B}$ is the Boltzmann constant, and $T_{\rm bath}$ is the bath temperature. To cool the qubit subsystem below $T_{\rm bath}$, a microwave magnetic flux of amplitude $A$ and frequency $\nu$ targets the $|0L\rangle \to |1R\rangle$ transition, driving the state-$|0L\rangle$ thermal population to state $|1R\rangle$, 