this area has been predominantly theoretical in nature, pointing out the conditions under which errors are introduced and methods to minimize their effects.[@Goorden03a; @Steffen03a; @Meier05a; @Amin06a; @Strauch06a; @Shevchenko07a] There has, however, been less direct experimental evidence confirming the validity of models upon which these predictions are based,[@Claudon04a; @Claudon07a; @Strauch06a; @Lucero08a] particularly with regard to leakage.

In this paper, we investigate the influence of the higher states of a phase qubit by examining the behavior of Rabi oscillations at high power, where their impact is greatest. In Sec. \[SPhase\], we describe the design of our qubit and a Hamiltonian that approximates its dynamics. Also discussed are various experimental details and the readout scheme that allows measurement of very small excited state populations. Section \[SStrong\] describes Rabi oscillations taken at a range of power and detuning from resonance, along with comparisons to a simple theory, while in Sec. \[SDecoherence\] we develop a more complete density-matrix model including the effects of decoherence and noise. Finally, Sec. \[SSummary\] contains a summary of the key points of our findings.

 SQUID Phase Qubit {#SPhase}
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![\[FDevice\]The dc SQUID phase qubit. (a) The qubit junction $J1$ (with critical current $I_{01}$ and capacitance $C_1$) is isolated from the 