all of the quantities in the equation are time dependent during a bias ramp.

This description of the qubit makes several simplifications. For one, we have ignored the quantum states of the isolation junction $J2$.[@Mitra06a] For typical bias conditions, we keep the current through this junction small and its first excited state is two to three times higher in energy than that of the qubit. However, the higher excited states of the qubit junction may come into resonance with states of the isolation junction. This approach further ignores the role of $J2$ in determining the bias conditions. Quantum mechanical simulations of the SQUID show that the current through the qubit junction weakly depends on its quantum state;[@Mitra06a] shifts in the qubit current are predicted to be less than 5 nA (much smaller than $I_{01} \approx 18\ {\ensuremath{\mu {\ensuremath{\mathrm{A}}}}}$) for our situation, so we have not considered this correction below. As a result, the junction parameters in [Eq. [(\[eqa\])]{}]{} need not be equal to the actual values for the qubit junction.

The simulations described in the sections that follow require values of [$\Gamma_{n}$]{}, [$\omega_{n m}$]{}, and [$\left< n \left| \, \gamma_1 \, \right|
                                    m \right>$]{}, all of which are specified by 