single current-biased junction,[@Fulton74a; @Leggett87a] $${\ensuremath{\mathcal{H}}}= \frac{4 E_C}{\hbar^2} p_1^2 - E_J {\ensuremath{\left( \cos \gamma_1
         + \frac{I_b - {\ensuremath{I_{r\!f}}}\cos {\ensuremath{\omega_{r\!f}}}t}{I_{01}} \gamma_1 \right)}}.
\label{eqa}$$ Here, $E_C = e^2 / 2 C_1$ and $E_J = I_{01} {\ensuremath{\Phi_0}}/ 2 \pi$ are the charging energy and Josephson coupling energy of the qubit junction, $p_1 = {\ensuremath{\left( {\ensuremath{\Phi_0}}/ 2 \pi \right)}}^2 C_1 \dot{\gamma_1}$ is the momentum conjugate to $\gamma_1$, and [$I_{r\!f}$]{} is the amplitude of a microwave drive current of frequency [$\omega_{r\!f}$]{}. In quantizing ${\ensuremath{\mathcal{H}}}$, $\gamma_1$ and $p_1$ become operators with ${\ensuremath{\left[ \gamma_1, p_1 \right]}} = i \hbar$.

The second term on the right-hand side of [Eq. [(\[eqa\])]{}]{} defines a one-dimensional tilted washboard potential $U$, sketched in Fig. \[FDevice\](b). Each well is characterized by the classical plasma frequency $\omega_p$ and barrier height $\Delta U$, which can also be expressed as the dimensionless quantity $N_s = \Delta U / \hbar \omega_p$. A single potential well supports roughly $N_s$ metastable states [$\left| n \right>$]{}, where the ground state [$\left| 0 \right>$]{} and first excited state [$\left| 1 \right>$]{} serve as the basis for quantum computation.[@Ramos01a] Motivated by this, the Hamiltonian can be expressed in a discrete representation as $${\ensuremath{\mathcal{H}}}_N = \sum_{n=0}^{N-1} \hbar {\ensuremath{\omega_{0 n}}} {\ensuremath{\left| n \right>}} {\ensuremath{\left< n 