\right|}}
         + \sum_{n, m = 0}^{N-1} \hbar {\ensuremath{\Omega_{n m}}} {\ensuremath{\left| n \right>}} {\ensuremath{\left< m \right|}} \cos {\ensuremath{\omega_{r\!f}}}t,
\label{eqc}$$ where $N$ is the number of states in the well being considered, $\hbar {\ensuremath{\omega_{n m}}}$ is the energy-level spacing between states [$\left| n \right>$]{} and [$\left| m \right>$]{} (tunable through $I_b$), and $${\ensuremath{\Omega_{n m}}} = \frac{{\ensuremath{\Phi_0}}}{2 \pi} \frac{{\ensuremath{I_{r\!f}}}}{\hbar} {\ensuremath{\left< n \left| \, \gamma_1 \, \right|
                                    m \right>}}
\label{eqf}$$ is a bare Rabi frequency.

The simplified Hamiltonian in [Eq. [(\[eqc\])]{}]{} neglects tunneling through the potential barrier,[@Larkin86a; @Leggett87a; @Chow88a; @Kopietz88a] a process corresponding to the SQUID spontaneously switching from the supercurrent to the voltage state; tunneling from [$\left| n \right>$]{} occurs at an escape rate [$\Gamma_{n}$]{}. As ${\ensuremath{\Gamma_{n+1}}} / {\ensuremath{\Gamma_{n}}} \sim 10^3$ for values of $I_b$ that we studied, tunneling can be exploited to perform state readout. Note also typically ${\ensuremath{\Gamma_{n}}} \ll {\ensuremath{\omega_{n, n+1}}}$ for the lowest few levels; for states near the barrier, this is not true. The total escape rate of the system to the voltage state is $$\Gamma = \frac{1}{\rho_{tot}} \sum_{n=0}^{N-1} \rho_{nn} {\ensuremath{\Gamma_{n}}},
\label{eqg}$$ where $\rho_{nn}$ is the occupation probability of state [$\left| n \right>$]{} and $\rho_{tot} = \sum \rho_{nn}$ is the probability of the system being in the supercurrent state; 