a persistent current $I_{\textrm{q}}$ with clockwise (counterclockwise) circulation. These two diabatic energy levels (Fig. \[fig:2\]A) have a separation $\varepsilon = 2I_{\textrm{q}} \delta f_{\textrm{dc}}$ linear in the flux detuning $\delta f_{\textrm{dc}} \equiv f_{\textrm{dc}} - \Phi_0/2$. Higher-excited states of the double-well potential (see Fig. \[fig:5\]C) will be considered in Sections \[sec:Cooling\] and \[sec:AS\].

The two-level Hamiltonian for the lowest two diabatic states is shown in Eq. \[Eq:Hamiltonian\]. At detuning $\delta f_{\textrm{dc}}=0$, the double-well potential is symmetric and the diabatic-state energies are degenerate. At this “degeneracy point,” resonant tunneling between the diabatic states opens an avoided level crossing of energy $\Delta$. Here, the qubit states are $\sigma_x$ eigenstates, corresponding to symmetric and anti-symmetric combinations of diabatic circulating-current states. Detuning the flux away from this point tilts the double well, allowing us to tune the eigenstates and eigenenergies of the artificial atom. Far from the degeneracy point the qubit states are approximately $\sigma_z$ eigenstates, the diabatic states with well-defined circulating current. The qubit is read out using a hysteretic DC SQUID (superconducting quantum interference device), a sensitive magnetometer that can distinguish the flux generated by circulating current states.

In addition to the static flux biases, the artificial atom is controlled and read out using 