  In photon recoil spectroscopy, signals are extracted from recoils imparted by
the spectroscopy light on the motion of trapped ions as demonstrated by C.
Hempel et al., Nature Photonics 7, 630 (2013) and Y. Wan et al., Nature
Communications 5, 3096 (2014). The method exploits the exquisite efficiency in
the detection of phonons achievable in ion crystals, and is thus particularly
suitable for species with broad non-cycling transitions where detection of
fluorescence photons is impractical. Here, we develop a theoretical model for
the description of photon recoil spectroscopy based on a Fokker-Planck equation
for the Wigner function of the phonon mode. Our model correctly explains
systematic shifts due to Doppler heating and cooling as observed in the
experiment. Furthermore, we investigate quantum metrological schemes for
enhancing the spectroscopic sensitivity based on the preparation and detection
of nonclassical states of the phonon mode.
