the Hamiltonian of [Eq. [(\[eqc\])]{}]{}, but are difficult to see in Fig. \[F031806DGtot\]. As shown by the solid lines in the figure, most of the features of the measured escape rate are captured by a multilevel density-matrix simulation which will be described in Sec. \[SDecoherence\].

![\[F031806DGtot\] Rabi oscillations in the escape rate $\Gamma$ were induced at $I_b = 17.746\ {\ensuremath{\mu {\ensuremath{\mathrm{A}}}}}$ by switching on a microwave current at $t = 0$ with a frequency of 6.2 GHz (resonant with the [$0 \rightarrow 1$]{} transition) and source powers [$P_S$]{} between $-12$ and $-32$ dBm, as labeled. The measurements were taken at 20 mK. The solid lines are from a five-level density-matrix simulation with $I_{01} = 17.930\ {\ensuremath{\mu {\ensuremath{\mathrm{A}}}}}$, $C_1 = 4.50\ {\ensuremath{{\ensuremath{\mathrm{pF}}}}}$, $T_1 = 17\ {\ensuremath{{\ensuremath{\mathrm{ns}}}}}$, and $T_\phi = 16\ {\ensuremath{{\ensuremath{\mathrm{ns}}}}}$.](031806DGtotv3.eps){width="3.0in"}

Detuning and Strong Field Effects {#SStrong}
=================================

We first model Rabi oscillations using the rotating wave approximation, in the absence of tunneling and dissipation. This provides a way of predicting the oscillation frequency for a wide range of experimental parameters.[@Steffen03a; @Claudon04a; @Meier05a; @Amin06a; @Strauch06a; @Claudon07b] In the rotating frame corresponding to the drive frequency ${\ensuremath{\omega_{r\!f}}}\approx {\ensuremath{\omega_{0 1}}}$, the Hamiltonian of [Eq. [(\[eqc\])]{}]{} simplifies to[@Strauch06a] $${\ensuremath{\mathcal{H}}}_N^{RW} = \sum_{n=0}^{N-1} \hbar \Delta_n {\ensuremath{\left| 