  Analyzing weak microwave signals in the GHz regime is a challenging task if
the signal level is very low and the photon energy widely undefined. A
superconducting qubit can detect signals in the low photon regime, but due to
its discrete level structure, it is only sensitive to photons of certain
energies. With a multi-level quantum system (qudit) in contrast, the unknown
signal frequency and amplitude can be deduced from the higher level AC Stark
shift. The measurement accuracy is given by the signal amplitude, its detuning
from the discrete qudit energy level structure and the anharmonicity. We
demonstrate an energy sensitivity in the order of $10^{-3}$ with a measurement
range of more than $1\,\mathrm{GHz}$. Here, using a transmon qubit, we
experimentally observe shifts in the transition frequencies involving up to
three excited levels. These shifts are in good agreement with an analytic
circuit model and master equation simulations. For large detunings, we find the
shifts to scale linearly with the power of the applied microwave drive.
Exploiting the effect, we demonstrated a power meter which makes it possible to
characterize the microwave transmission from source to sample.
