  Trapped ions are sensitive detectors of weak forces and electric fields that
excite ion motion. Here measurements of the center-of-mass motion of a
trapped-ion crystal that are phase-coherent with an applied weak external force
are reported. These experiments are conducted far from the trap motional
frequency on a two-dimensional trapped-ion crystal of approximately 100 ions,
and determine the fundamental measurement imprecision of our protocol free from
noise associated with the center-of-mass mode. The driven sinusoidal
displacement of the crystal is detected by coupling the ion crystal motion to
the internal spin-degree of freedom of the ions using an oscillating
spin-dependent optical dipole force. The resulting induced spin-precession is
proportional to the displacement amplitude of the crystal, and is measured with
near-projection-noise-limited resolution. A $49\,$pm displacement is detected
with a single measurement signal-to-noise ratio of 1, which is an
order-of-magnitude improvement over prior phase-incoherent experiments. This
displacement amplitude is $40$ times smaller than the zero-point fluctuations.
With our repetition rate, a $8.4\,$pm$/\sqrt{\mathrm{Hz}}$ displacement
sensitivity is achieved, which implies
$12\,$yN$/\mathrm{ion}/\sqrt{\mathrm{Hz}}$ and
$77\,\mu$V$/$m$/\sqrt{\mathrm{Hz}}$ sensitivities to forces and electric
fields, respectively. This displacement sensitivity, when applied on-resonance
with the center-of-mass mode, indicates the possibility of weak force and
electric field detection below $10^{-3}\,$yN/ion and $1\,$nV/m, respectively.
