  The use of special quantum states to achieve sensitivities below the limits
established by classically behaving states has enjoyed immense success since
its inception. In bosonic interferometers, squeezed states, number states and
cat states have been implemented on various platforms and have demonstrated
improved measurement precision over interferometers based on coherent states.
Another metrologically useful state is an equal superposition of two
eigenstates with maximally different energies; this state ideally reaches the
full interferometric sensitivity allowed by quantum mechanics. By leveraging
improvements to our apparatus made primarily to reach higher operation
fidelities in quantum information processing, we extend a technique to create
number states up to $n=100$ and to generate superpositions of a harmonic
oscillator ground state and a number state of the form
$\textstyle{\frac{1}{\sqrt{2}}}(\lvert 0\rangle+\lvert n\rangle)$ with $n$ up
to 18 in the motion of a single trapped ion. While experimental imperfections
prevent us from reaching the ideal Heisenberg limit, we observe enhanced
sensitivity to changes in the oscillator frequency that initially increases
linearly with $n$, with maximal value at $n=12$ where we observe 3.2(2) dB
higher sensitivity compared to an ideal measurement on a coherent state with
the same average occupation number. The quantum advantage from using
number-state superpositions can be leveraged towards precision measurements on
any harmonic oscillator system; here it enables us to track the average
fractional frequency of oscillation of a single trapped ion to approximately
2.6 $\times$ 10$^{-6}$ in 5 s. Such measurements should provide improved
characterization of imperfections and noise on trapping potentials, which can
lead to motional decoherence, a leading source of error in quantum information
processing with trapped ions.
