Photon-recoil spectroscopy: Systematic shifts and nonclassical enhancements

In photon-recoil spectroscopy, signals are extracted from recoils imparted by the spectroscopy light on the motion of trapped ions as demonstrated by Hempel et al. [C. Hempel et al., Nat. Photon. 7, 630 (2013)] and Wan et al. [Y. Wan et al., Nat. Commun. 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 noncycling 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.