Measurement of the Excited State Lifetime of a Cooper-Pair Box

We have used a radio frequency superconducting single electron transistor (rf-SET) biased around the double Josephson quasiparticle peak to measure the lifetime T$_{1}$ of the excited state of an Al/AlO$_{\mbox{x}}$/Al Cooper-pair box (CPB) qubit. The CPB had a charging energy E$_{C}$/k$_{B}$ = 0.78 K and a maximum Josephson coupling energy E$_{J}$/k$_{B}$ = 0.70 K and all measurements were made at about 40 mK. T$_{1}$ was found by sending a pulse of microwaves to the gate of the CPB and then using the rf-SET to observe the decay rate of the charge signal on the CPB. Near the degeneracy point of the CPB, we observed T$_{1}$ of approximately 100 ns, which was near the limit of the rf-SET bandwidth. As we move away from the degeneracy point, T$_{1}$ varies, reaching a maximum of approximately 400 ns. We examine whether these changes in T$_{1}$ are commensurate with the quantum noise spectral density from the rf-SET.