Levitated optomechanics is showing potential for precise force measurements. Here, we report a case study, to show experimentally the capacity of such a force sensor. Using an electric field as a tool to detect a Coulomb force applied onto a levitated nanosphere. We experimentally observe the spatial displacement of up to 6.6 nm of the levitated nanosphere by imposing a DC field. We further apply an AC field and demonstrate resonant enhancement of force sensing when a driving frequency, ωAC , and the frequency of the levitated mechanical oscillator, ω0, converge. We directly measure a force of 3.0 ± 1.5 × 10 −20 N with 10 second integration time, at a centre of mass temperature of 3 K and at a pressure of 1.6 × 10 −5 mbar.The ability to detect forces with increasing sensitivity, is of paramount importance for many fields of study, from detecting gravitational waves [1] to molecular force microscopy of cell structures and their dynamics [2]. In the case of a mechanical oscillator, the force sensitivity limit arises from the classical thermal noise, as given by,Where, k b is the Boltzmann constant, T is the temperature of the thermal environment, m, the mass of the object, ω 0 is the oscillator angular frequency, Q m = ω 0 /Γ 0 is the mechanical quality factor and Γ 0 is the damping factor. In recent decades, systems, such as cold-atoms traps, have pushed the boundaries of force sensitivities down to 1 × 10 [3,35]. The control of charges on nanoparticles is essential for experiments to prepare non-classical states of motion of the particle [30,36]. In addition, force detection at 1.63 × 10 −18 N/ √ Hz in levitated nanospheres has already been demonstrated [20] by experiment.Here, we take a detailed look at the interaction of an optically levitated dielectric charged particle with an external electric field as a case study for force sensing. We measure the effect of the Coulomb interaction on the motion of a single nanoparticle, at high vacuum (10 −5 mbar) by applying a DC and an AC electric field to a metallic needle positioned near the trapped particle. These particles can carry multiple elementary electric charges (e = 1.6 × 10 −19 C), and we use the Coulomb interaction to determine the number of elementary charges attached to the particle.The charge at the needle tip, q t , for a given applied voltage is according to Gauss's Law, s E·ds t = qt 0 , where s t is the surface of the needle tip, 0 is the vacuum permittivity, and E is the electric field. The electric field at any point in a potential, V , is given by −∇V = E. If, we approximate the needle tip as a sphere, of radius, r t , arXiv:1706.09774v3 [quant-ph]
