independently control the currents in the two arms of the resulting dc SQUID, a current source $I_f$ applies a flux $\Phi_a$ to the SQUID loop through mutual inductance $M$. Good isolation of the qubit junction is obtained when $L_1 / M \gg 1$ and $L_1 / {\ensuremath{\left( L_2 + L_{J2} \right)}} \gg 1$, where $L_{J2}$ is the Josephson inductance of the isolation junction.[@Martinis02a]

For arbitrary values of the bias current $I_b$ and flux current $I_f$, the dynamics of a dc SQUID can be described by 2 degrees of freedom corresponding to the phase differences across each of the junctions. We, however, operate the device by increasing $I_b$ by $\Delta I_b$ while simultaneously increasing $I_f$ by $L_1 \Delta I_b / M$. This nominally keeps the total current through the isolation junction $J2$ near zero, so that the qubit junction current is roughly $I_b$. Furthermore, one obtains a weak dynamical coupling between the junctions by choosing $L_1$ to be large and biasing the SQUID so that the two junctions are well out of resonance with each other.[@Palomaki06a] In this case, the dynamics of the phase difference $\gamma_1$ across the qubit junction are governed to a good approximation[@Mitra06a] by the Hamiltonian of a 