Dephasing Measurements of a Cooper-pair box

We present data on the dephasing properties of our Al/AlO$_{\mbox{x}}$/Al Cooper-pair box (CPB) qubit. The CPB had a charging energy $E_{C}/h$ = 6.25 GHz and a maximum $E_{J}/h$ = 19 GHz which was decreased by an external magnetic field to an effective $E_{J}/h$ of 6.1 GHz. The qubit was capacitively coupled to a lumped element microwave resonator ($f_{0} = 5.446$ GHz, $Q_{L} = 1.8\times10^{4}$) which was in turn coupled to a transmission line. To manipulate the qubit, a microwave pulse at 6.1 GHz was sent to the transmission line. The state of the qubit was then measured by sending a second microwave pulse at $f_{0}$ and measuring the amplitude and phase of the transmitted power. We observed Rabi oscillations with Rabi frequencies from 1.94 to 5.32 MHz decay with time constants in the range T' = 0.6 to 1.6 $\mu$s. We measured an inhomogeneous dephasing time ($T_{2}${*}) of 322 ns by performing a Ramsey fringe experiment. Assuming $1/f$ charge noise is the dominant dephasing mechanism we extracted a $1/f$ charge noise amplitude of 1.6$\times$10$^{-3} e/\sqrt{Hz}$ at 1 Hz.