  We present a technique to measure the amplitude of a center-of-mass (COM)
motion of a two-dimensional ion crystal of $\sim$100 ions. By sensing motion at
frequencies far from the COM resonance frequency, we experimentally determine
the technique's measurement imprecision. We resolve amplitudes as small as 50
pm, 40 times smaller than the COM mode zero-point fluctuations. The technique
employs a spin-dependent, optical-dipole force to couple the mechanical
oscillation to the electron spins of the trapped ions, enabling a measurement
of one quadrature of the COM motion through a readout of the spin state. We
demonstrate sensitivity limits set by spin projection noise and spin
decoherence due to off-resonant light scattering. When performed on resonance
with the COM mode frequency, the technique demonstrated here can enable the
detection of extremely weak forces ($< \,$1 yN) and electric fields ($< \,$1
nV/m), providing an opportunity to probe quantum sensing limits and search for
physics beyond the standard model.
