  We sense the motion of a trapped atomic ion using a sequence of
state-dependent ultrafast momentum kicks. We use this atom interferometer to
characterize a nearly-pure quantum state with $n=1$ phonon and accurately
measure thermal states ranging from near the zero-point energy to $\bar{n}\sim
10^4$, with the possibility of extending at least 100 times higher in energy.
The complete energy range of this method spans from the ground state to far
outside of the Lamb-Dicke regime, where atomic motion is greater than the
optical wavelength. Apart from thermometry, these interferometric techniques
are useful for characterizing ultrafast entangling gates between multiple
trapped ions.
