  Coherently displaced harmonic oscillator number states of a harmonically
bound ion can be coupled to two internal states of the ion by a laser-induced
motional sideband interaction. The internal states can subsequently be read out
in a projective measurement via state-dependent fluorescence, with near-unit
fidelity. This leads to a rich set of line shapes when recording the
internal-state excitation probability after a sideband excitation, as a
function of the frequency detuning of the displacement drive with respect to
the ion's motional frequency. We precisely characterize the coherent
displacement based on the resulting line shapes, which exhibit sharp features
that are useful for oscillator frequency determination from the single quantum
regime up to very large coherent states with average occupation numbers of
several hundred. We also introduce a technique based on multiple coherent
displacements and free precession for characterizing noise on the trapping
potential in the frequency range of 500 Hz to 400 kHz. Signals from the ion are
directly used to find and eliminate sources of technical noise in this
typically unaccessed part of the spectrum.
