Sensing Static Forces with Free-Falling Nanoparticles

Miniaturized mechanical sensors rely on resonant operation schemes, unsuited to detect static forces. We demonstrate a nanomechanical sensor for static forces based on an optically trapped nanoparticle in vacuum. Our technique relies on an off-resonant interaction of the particle with a weak static force, and a resonant readout of the displacement caused by this interaction. We demonstrate a sensitivity of 10 aN to static gravitational and electric forces. Our work provides a tool for the closer investigation of short-range forces, and marks an important step towards the realization of matter-wave interferometry with macroscopic objects.

Figure 1

(a) To initialize the system, we trap a nanoparticle with low energy $E_0$ in an optical potential (blue). (b) We deactivate the trapping potential for a time $\tau$, during which the particle gets displaced by the static force $F$. (c) Upon reactivation of the optical trap, the particle gains potential energy due to its displacement from the trap center and oscillates with a higher amplitude than in (a). (d) Experimental setup. The nanoparticle is levitated in the focus of a laser beam inside a vacuum chamber. Gravity acts along the y axis. We switch the optical trap off with an electro- and acousto-optical modulator (EOM, AOM), measure the particle position with a balanced detector, and feedback cool the c.m. energy by modulating the trapping beam. Applying a voltage between the objective and the holder of the collection lens allows us to create a static electric field exerting a static force along the z axis on a charged particle.

Figure 2

(a) Timing scheme of a single free-fall cycle [cf. Figs. 1]. For $t<0$, the trapping potential is on and the feedback (FB) cools the particle’s c.m. motion. From $t=0$ to $t=\tau =100\mu \mathrm{s}$, we deactivate the optical trap and the feedback such that the particle can freely interact with the static force. At time $t=\tau$, the trap is reactivated while the feedback remains off. We plot the recorded motion along the $y$ axis (b) and the $x$ axis (c) in black. For $t>\tau$, the particle undergoes sinusoidal oscillations (orange and blue fit) with a larger amplitude along the $y$ direction. (d) Average oscillation energy $\overline{E}$ after free fall in the $x$ direction for varying fall durations $\tau$ (error bars are smaller than the symbol size). (e) Along the $y$ axis, gravity accelerates the particle. For both axes, we fit $\overline{E}={\beta }_0+{\beta }_2{\tau }^2+{\beta }_4{\tau }^4$ (solid line) and show the ${\beta }_2{\tau }^2$ term (dotted line), and the ${\beta }_4{\tau }^4$ term (dashed line). The latter only contributes for the $y$ axis measurement, and is a signature of the gravitational acceleration.

Figure 3

Calculated phase-space distributions. (a) Initial probability distribution in phase space, corresponding to time $t=0$ in Figs. 2, shown for a thermal state with a c.m. energy of $E_0/k_B=50\mathrm{mK}$. (b) Directly after a free evolution of duration $\tau =100\mu \mathrm{s}$ with deactivated trapping potential and in the absence of any force, the distribution is spread along the position axis due to the initial velocity of the particle. (c) Phase-space distribution after the evolution of duration $\tau =100\mu \mathrm{s}$ in the presence of a static force $F=-20\mathrm{aN}$. Compared to (b), the distribution is displaced by $\mathrm{\Delta }y=F{\tau }^2/(2m)=-49\mathrm{nm}$ and $\mathrm{\Delta }\dot{y}=F\tau /m=-0.98\mathrm{nm}/\mu \mathrm{s}$.

Figure 4

(a) Mean energy in the $z$ direction as a function of interaction time $\tau$ at different electric field strengths (squares and triangles). We fit $\overline{E}={\beta }_0+{\beta }_2{\tau }^2+{\beta }_4{\tau }^4$ (solid lines) and indicate the ${\beta }_4{\tau }^4$ component of the fit (dashed) for the data taken with a charged particle. The energy of an uncharged particle (gray points) scales with ${\tau }^2$ (dotted line, the initial energy is higher than in the charged case). Inset: Sketch of capacitor formed by objective and collection lens holder (metal gray, glass blue), and field lines (red). (b) Fitting parameters ${\beta }_4$ from (a) with ${\beta }_4(\mathcal{E})=q^2(\mathcal{E}+{\mathcal{E}}_{\mathrm{res}}{)}^2{\mathrm{\Omega }}_0^2/(8m)$ (black). The deduced fitting parameter is the residual electric field strength ${\mathcal{E}}_{\mathrm{res}}=452(13)\mathrm{V}/\mathrm{m}$.