Quantum State Tomography of a Cooper-pair Box

A 4-8 GHz microwave pulse shaping system with 3 ns Gaussian pulse rise time, arbitrary pulse envelope and phase control has been implemented. The system utilizes a two-channel 1 GSa/s DAC board\footnote{Designed by J. Martinis at UCSB and fabricated by HSCC.} to supply control voltages to an IQ mixer. The signals to the mixer have been optimized to obtain an on-off ratio of $>85\mbox{ dB}$ and phase deviations $<5\%$. The setup has been used to manipulate an $Al/AlO_{x}/Al$ Cooper-pair box (CPB) qubit coupled to a lumped-element microwave resonator ($f_0 = 5.446 \mbox{ GHz}$). The CPB has a charging energy $E_C/h = 6.25\mbox{ GHz}$ and a maximum $E_J/h = 19\mbox{ GHz}$ which was decreased to an effective $E_J/h = 6.1\mbox{ GHz}$ by an external magnetic field. By measuring the microwave transmission at $f_0$ in a pulsed-probe scheme, we perform a dispersive readout of the qubit. We present tomography data on the $\vert g \rangle$, $\vert e \rangle$, $(\vert g \rangle + \vert e \rangle)/\sqrt{2}$ and $(\vert g \rangle + i \vert e \rangle)/\sqrt{2}$ states. We find good agreement with theory, confirming that we have achieved the desired microwave pulse control.