f_{\textrm{dc}}$=0. This plot constitutes a cumulative switching current distribution of the SQUID for each $\delta f_{\textrm{dc}}$ value. Additionally, a 3-$\mu$s RF pulse at 1.2 GHz is applied to the qubit, and resonant transitions can be observed as fingers extending down (up) from state $|L \rangle$ ($|R \rangle$) when $\textrm{n}\times$1.2 GHz becomes resonant with the energy-level separation. A best-estimator (dashed-dotted line) can be determined to provide the best statistical discrimination between states $|L \rangle$ and $|R \rangle$. The resulting qubit step with its saturated n-photon resonances along the best estimator line is shown in Fig. \[fig:1\]E.

The device used in this work was fabricated at Lincoln Laboratory using a fully-planarized niobium trilayer process and optical lithography. It has a critical current density $J_{\rm c} \approx 160 \,{\rm A/cm^2}$, and the characteristic Josephson and charging energies are $E_{\textrm{J}} \approx (2\pi\hbar)300\,{\rm{GHz}}$ $E_{\textrm C} \approx (2\pi\hbar)0.65\,{\textrm{GHz}}$ respectively. The ratio of the qubit Josephson junction (JJ) areas is $\alpha \approx 0.84$, and $\Delta \equiv \Delta_{0,0} \approx (2\pi\hbar)10\,{\textrm{MHz}}$. Although dependent on the flux detuning, the approximate values for coherence times are: interwell relaxation time $T_1 \sim 100 \,\mu \textrm{s}$, intrawell relaxation time $T_1 \sim 0.05 \,\mu \textrm{s}$, homogeneous transverse decay time $T_2 \sim 20 \, \textrm{ns}$, 