Analyzing weak microwave signals in the GHz regime is a challenging task ifthe signal level is very low and the photon energy widely undefined. Due to itsdiscrete level structure, a superconducting qubit is only sensitive to photonsof certain energies. With a multi-level quantum system (qudit) in contrast, theunknown photon frequency can be deduced from the higher level AC Stark shift.The measurement accuracy is given by the signal amplitude, its detuning fromthe discrete qudit energy level structure and the anharmonicity. We demonstratean energy sensitivity in the order of $10^{-4}$ with a measurement range of 1GHz. Here, using a transmon qubit, we experimentally observe shifts in thetransition frequencies involving up to three excited levels. These shifts arein good agreement with an analytic circuit model and master equationsimulations. For large detunings, we find the shifts to scale linearly with thepower of the applied microwave drive.
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