$I_b$, $I_{01}$, and $C_1$. In the absence of dissipation, analytical expressions for these quantities can be obtained by applying perturbation theory to the cubic approximation of the tilted washboard potential.[@Strauch04a] For the escape rates and energy levels, we instead use numerical solutions to the exact potential, which are more reliable for states near the top of the barrier. These solutions come from two methods that give consistent results: (i) solving the Schrödinger equation with transmission boundary conditions[@Shibata91a] and (ii) complex scaling.[@Moiseyev98a]

We performed experiments on a dc SQUID phase qubit fabricated by Hypres, Inc.[@NIST] using a $30\ {\ensuremath{{\ensuremath{\mathrm{A}}} / {\ensuremath{\mathrm{cm}}}^2}}$ Nb/AlO$_\text{x}$/Nb trilayer process \[see Fig. \[FDevice\](c)\]. The qubit and isolation junctions had areas of $10 \times 10~{\ensuremath{\mu {\ensuremath{\mathrm{m}}}}}^2$ and $7 \times 7~{\ensuremath{\mu {\ensuremath{\mathrm{m}}}}}^2$, respectively, and the inductances were roughly $L_1 = 3.4\ {\ensuremath{{\ensuremath{\mathrm{nH}}}}}$, $L_2 = 30\ {\ensuremath{{\ensuremath{\mathrm{pH}}}}}$, and $M = 13.4\ {\ensuremath{{\ensuremath{\mathrm{pH}}}}}$.[@Palomaki07a] The error in these and other fit parameters reported in this paper is about one unit in the least significant digit. We have not attempted a thorough analysis of the uncertainties in all the parameters due to the complexities of the nonlinear functions involved. The microwave current [$I_{r\!f}$]{} was carried inside the refrigerator on a single length of 