n \right>}} {\ensuremath{\left< n \right|}}
             + \frac{1}{2} \sum_{\substack{n,m=0 \\ n \neq m}}^{N-1}
               \hbar \Omega_{nm}^\prime {\ensuremath{\left| n \right>}} {\ensuremath{\left< m \right|}},
\label{eqd}$$ where $\Delta_n = {\ensuremath{\omega_{0 n}}} - n {\ensuremath{\omega_{r\!f}}}$ and $$\Omega_{nm}^\prime = {\ensuremath{\Omega_{n m}}} \sum_{s=0}^1 J_{m-n+2s-1}
                       {\ensuremath{\left( \frac{{\ensuremath{\Omega_{n n}}} - {\ensuremath{\Omega_{m m}}}}{{\ensuremath{\omega_{r\!f}}}} \right)}}$$ for $n < m$ and $\Omega_{mn}^\prime = \Omega_{nm}^\prime$. Here, $J_n {\ensuremath{\left( x \right)}}$ is the $n$th order Bessel function and $\Omega_{nm}$ is defined by [Eq. [(\[eqf\])]{}]{}.

Differences between the $N$ eigenvalues of this Hamiltonian specify effective multilevel Rabi frequencies or modes of the system. We find it convenient to label these frequency differences by the states [$\left| n \right>$]{} with the largest weight in the two corresponding eigenfunctions. For the parameter regime of interest here, the differences can be uniquely classified by two states, which we denote by [$\Omega_{R,n m}$]{}. For example, [$\Omega_{R,0 2}$]{} denotes the Rabi oscillation frequency between eigenstates that describe a two-photon transition between [$\left| 0 \right>$]{} and [$\left| 2 \right>$]{}. While approximate analytical solutions can be obtained for a three-level junction system,[@Strauch06a] we numerically found the eigenvalues of the rotating wave Hamiltonian for systems with up to seven levels in the simulations that follow.

![\[F031806Dfosc\] Rabi oscillation frequency [$\Omega_{R,0 1}$]{} at fixed bias 