The state of the qutrit was obtained by three-level quantum tomography, where the diagonal elements of the density matrix were calculated from the averaged IQ traces of the cavity response (44). The measured traceEmbedded Image(9)is a linear combination of calibration traces corresponding to states |0〉, |1〉, and |2〉 with weight factors p0, p1, and p2, which give the occupation probability of each state. Here, τ is the time from the beginning of the measurement pulse. Using the least squares fit of the calibration traces to the measured trace, we can extract the most likely occupation probabilities for the three-level system.

The calibration traces inevitably include the effect of relaxation, which, if left uncompensated, can lead to an artificial overestimation of the state population in both STIRAP and saSTIRAP. However, since we know the relaxation rates, we can correct for this effect by modifying the calibration trajectories to include some contribution from the lower states, described by errors ζij with i < j. The measured trajectory rj is then given byEmbedded Image(10)with Embedded Image describing the unknown ideal responses of state |i〉. From the above equation, the ideal responses can be solved iteratively, yieldingEmbedded Image(11)

We used ζ01 = 0.01, ζ12 = 0.01, and ζ02 = 0.02, which are obtained by comparing a reference Rabi experiment to a corresponding simulation with known energy relaxation rates.

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