Nanomechanical sensor resolving impulsive forces below its zero-point fluctuations

TL;DR

The paper tackles the quantum-limited sensitivity of mechanical transducers to impulsive forces by introducing a coherent amplification protocol based on reversible squeezing of the transducer's motion. By temporarily lowering the trap frequency from $\Omega$ to $\Omega/r$ to squeeze momentum and then anti-squeezing to convert a momentum kick into an amplified position displacement $\Delta Q = r\,\Delta P$, the authors demonstrate impulsive-force detection below the zero-point fluctuations $p_{\text{zp}}$ on an optically levitated nanoparticle. They achieve a minimum detectable impulse of $6.9 \pm 0.8$ keV/$c$, about $0.6^{+0.6}_{-0.4}$ dB below $p_{\text{zp}}$ and $2.1$ dB below the ideal continuous-sensing limit, with performance limited by squeezing-induced backaction and photon recoil at higher $r$. The work paves the way for enhanced quantum sensing in massive mechanical systems and hints at applications in detecting new physics or rare nuclear processes, with future improvements possible via hybrid traps and alternative squeezing methods.

Abstract

The sensitivity of a mechanical transducer is ultimately limited by its inherent quantum fluctuations. Here, we use an optically levitated nanoparticle to measure impulsive forces smaller than the particle's zero-point momentum uncertainty. Our approach relies on reversibly squeezing the levitated particle's center-of-mass motion to coherently amplify the perturbation. We demonstrate resolving single impulsive-force kicks as small as 6.9 keV/c, a value 0.6 dB below the sensor's zero-point value.

Figures (5)

• Figure 1: (a) Conventional impulsive-force sensing. Left: A harmonic oscillator is initialized to a Gaussian state centered in phase-space with momentum uncertainty $\sigma_{P,i}$ (red). Right: An impulsive force translates the state by $\Delta P$ along the momentum axis. Reading out the displaced state adds estimation uncertainty $\sigma_{P,f}$ (blue). (b) Coherent amplification. Left: The harmonic oscillator is initialized as in (a). The momentum uncertainty is squeezed to $\sigma_{P,i}/r$ (purple ellipse). Center: The impulsive force displaces the state by $\Delta P$ along the momentum axis. Right: Anti-squeezing turns the momentum displacement $\Delta P$ into a position displacement $\Delta Q=r\Delta P$. (c) Modulation sequence for oscillator resonance frequency. The frequency is temporarily stepped from $\Omega$ to the reduced value $\Omega/r$ to squeeze and subsequently anti-squeeze the oscillator. The impulsive force acts at the time of maximum momentum squeezing.
• Figure 2: Experimental setup. A focused laser generates an optical trap for a dielectric nanoparticle in ultra-high vacuum (UHV). The backscattered light is mixed with a local oscillator (LO) to detect the particle motion along the $z$ axis. This position signal is used for feedback (FB) cooling. The optical trapping potential is modulated with an acousto-optic modulator (AOM). An impulsive force $\Delta P$ is imparted by applying a voltage pulse of duration $\tau$ and amplitude $U_p$ to metallic electrodes surrounding the nanoparticle.
• Figure 3: Conventional protocol (a--d) vs. coherent mechanical amplification (e--h). (a) Gray: Unfiltered time trace of measured position $Q(t)$. An impulsive force (duration $\tau=1000$ ns, illustrated in blue, not to scale) is applied at $t=0$. Red: Filtered timetrace. (b) State-space distribution right after application of the impulsive force for 200 realizations of the protocol as shown in (a). Dashed ellipse indicates the covariance matrix of the distribution. The displacement of the state by $\Delta P$ is visible. (c, d) Same as (a, b), but with $\tau=100$ ns. The displacement of the state is not discernible against the uncertainty. (e) Time trace of a coherent amplification protocol with squeezing factor $r=\sqrt{4}$. The impulsive force ($\tau=100~$ns) has the same strength as in (c, d). (f) State-space distribution obtained from 200 repetitions of the protocol shown in (e). A slight displacement of the state along the position axis is discernible. (g, h) Same as (e, f), but with squeezing factor $r=\sqrt{12}$. The position displacement in phase-space is visibly amplified by the squeezing factor as compared to (d).
• Figure 4: (a) Phase-space displacement $\Delta Q$ as a function of pulse duration $\tau$ for three different squeezing ratios $r$. Solid lines: Linear fits $\Delta Q = k(r)\,\tau$. Error bars: $1\sigma$ uncertainty obtained via error propagation. (b) Slope $k(r)$ as a function of squeezing ratio $r$ extracted from data as shown in (a). Dashed line indicates $k(r) = k_1\,r$. Shaded area: $1\sigma$ confidence interval of $k_1$. (c) Total measured position standard deviation $\sigma_{Q,\text{tot}}$ as a function of $r$. The slight increase with $r$ is due to measurement backaction. Colored datapoints in (b) and (c) correspond to datasets shown in (a). Shaded area: $1\sigma$ confidence interval of the model supplement.
• Figure 5: Minimum detectable impulsive force $\Delta P_{\text{min}}$ as a function of squeezing ratio $r$. Red dashed line: Limit of the conventional protocol for an otherwise ideal system. The label $\sqrt{2}p_\text{zp}$ is expressed in absolute units. Grey dashed line: Value of the momentum zero-point fluctuations of the levitated oscillator (label in absolute units). Error bars: $1\sigma$ uncertainty obtained from error propagation. Shaded area: $1\sigma$ confidence interval of the model supplement.