and [$\omega_{2 3}$]{} (dotted) obtained with the effective parameters.[@Strauch06a] In addition, two-photon transitions between these levels occurred at high power at the expected frequencies. Multiphoton transitions between the ground and first excited states of a variety of junction qubits have previously been observed and characterized.[@Nakamura01a; @Wallraff03a; @Saito04a] Over the course of three months (during which the refrigerator remained near its base temperature), the junction parameters found from the spectral fits varied by roughly 1%. Thus, we repeated the spectroscopy measurement often to obtain the values of $I_{01}$ and $C_1$ needed for the simulations; these values are listed in the caption of each figure.

The open circles of Fig. \[FEnergyGamma\](b) show the measured background escape rate in the absence of microwaves. The solid line indicates the theoretical value of [$\Gamma_{0}$]{}, calculated with the values of $I_{01}$ and $C_1$ extracted from the spectrum. At low bias current, the measured escape rate exceeds [$\Gamma_{0}$]{}, suggesting the presence of excited state population even though the refrigerator was at 20 mK. Based on experiments described elsewhere,[@Palomaki07a] we believe that features such as the peak near $I_b = 17.73\ {\ensuremath{\mu {\ensuremath{\mathrm{A}}}}}$ are due to population in [$\left| 2 \right>$]{} generated when noise on the leads at 