Minimum Requirements for Feedback Enhanced Force Sensing

The problem of estimating an unknown force driving a linear oscillator is revisited. When using linear measurement, feedback is often cited as a mechanism to enhance bandwidth, sensitivity or resolution. We show that as long as the oscillator dynamics are known, there exists a real-time estimation strategy that reproduces the same measurement record as any arbitrary feedback protocol. Consequently some form of nonlinearity is required to gain any advantage beyond estimation alone. This result holds true in both quantum and classical systems, with nonstationary forces and feedback, and in the general case of non-Gaussian and correlated noise. Recently, feedback enhanced incoherent force resolution has been demonstrated [E. Gavartin, P. Verlot, and T. J. Kippenberg, Nat. Nano. 7, 509 (2012)], with the enhancement attributed to a feedback induced modification of the mechanical susceptibility. As a proof-of-principle, we experimentally reproduce this result through straightforward filtering.

Figure 1

(a) Conceptual diagram comparing optomechanical force sensing with feedback and filtering. (b) Experimental schematic. Red (dark gray): fiber interferometer; solid green (light gray): electrical components for feedback stabilization; dashed green: electrical signal applied to the microtoroid for application of the incoherent force and feedback. FPC: fiber polarization controller. MZM: Mach-Zender Modulator. PM: phase modulator. PI: proportional-integral controller.

Figure 2

(a) Displacement spectrum at room temperature [dark gray (red)] and with filtering [light gray (green)] at gain $2$, $8$, $34$, $72$, and $150$. Inset top: displacement spectrum with feedback cooling [black (blue)] and filtering [light gray (green)] with gain $1$. Inset bottom: percentage difference between filtered and cooled spectrum with equivalent gain. (b) Force resolution as a function of averaging duration for thermal [dark gray (red)], feedback cooled [black (blue)], and filtered spectra [light gray (green)] at gain $1$, $2.4$, $5$, and $10$. Solid line (red): fit to inverse quartic dependence of the force resolution without filtering or feedback. (c) Force resolution versus filter gain after $1\mathrm{ms}$ of averaging (green circles) showing good agreement to theory (dashed line).

Figure 3

(a) Normalized energy estimate at room temperature [light gray (red)], and with addition of incoherent driving [dark gray (green)]. Solid lines: mean; dashed lines: one standard deviation error bounds. (b) Averaging time required to resolve incoherent signal versus filter gain (green circles) showing good agreement to theory (dashed line).