Quantum Lock-in Force Sensing using Optical Clock Doppler Velocimetry

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 $5.3\times10^{-19}\frac{\mathrm{N}}{\sqrt{\mathrm{Hz}}}$ .