  Force sensors are at the heart of different technologies such as atomic force
microscopy or inertial sensing \cite{RMPforce2003, Rugar2004, YazdiIEEE}. These
sensors often rely on the measurement of the displacement amplitude of
mechanical oscillators under applied force. Examples for such mechanical
oscillators include micro-fabricated cantilevers \cite{YazdiIEEE}, carbon
nanotubes \cite{NanotubeForce} as well as single trapped ions \cite{Bollinger,
Udem} . The best sensitivity is typically achieved when the force is
alternating at the mechanical resonance frequency of the oscillator thus
increasing its response by the mechanical quality factor. The measurement of
low-frequency forces, that are below resonance, is a more difficult task as the
resulting oscillation amplitudes are significantly lower. Here we use a single
trapped $^{88}Sr^{+}$ ion as a force sensor. The ion is trapped in a linear
harmonic trap, and is electrically driven at a frequency much lower than the
trap resonance frequency. To be able to measure the small amplitude of motion
we combine two powerful techniques. The force magnitude is determined by the
measured periodic Doppler shift of an atomic optical clock transition and the
Quantum Lock-in technique is used to coherently accumulate the phases acquired
during different force half-cycles. We demonstrate force detection both when
the oscillating force is phase-synchronized with the quantum lock-in sequence
and when it is asynchronous and report frequency force detection sensitivity as
low as $5.3\times10^{-19}\frac{\mathrm{N}}{\sqrt{\mathrm{Hz}}}$ .
