Bayesian Information Engine that Optimally Exploits Noisy Measurements

We have experimentally realized an information engine consisting of an optically trapped, heavy bead in water. The device raises the trap center after a favorable “up” thermal fluctuation, thereby increasing the bead’s average gravitational potential energy. In the presence of measurement noise, poor feedback decisions degrade its performance; below a critical signal-to-noise ratio, the engine shows a phase transition and cannot store any gravitational energy. However, using Bayesian estimates of the bead’s position to make feedback decisions can extract gravitational energy at all measurement noise strengths and has maximum performance benefit at the critical signal-to-noise ratio.

Figure 1

Schematic information engine. (a) Noisy detector measures position $y$ of the bead, actually located at $x$. Ratchet based on either (b) noisy position measurement $y$ or (c) Bayesian position estimate $\widehat{x}$ (blue dashed circle).

Figure 2

Tuning the feedback gain $\alpha$ to set trap power $\dot{W}=0$. (a) Trap power for naive (red) and Bayesian (blue) information engines at fixed $\mathrm{SNR}=11.$ (b) Measured bead ($y(t)$, red) and trap ($\lambda (t)$, black) trajectories for the naive information engine at $\mathrm{SNR}=11$. (c) Measured bead ($y(t)$, light red), filtered bead estimate ($\widehat{x}(t)$, blue), and trap ($\lambda (t)$, black) trajectories for the Bayesian information engine at $\mathrm{SNR}=2$. (b) and (c) have equal scale bars and satisfy $\dot{W}=0$. (d) Critical feedback gain ${\alpha }^{*}$ and (e) corresponding input trap power, for naive (red) and Bayesian (blue) information engines. Hollow red markers denote SNRs for which ${\alpha }^{*}>0$ could not be found using the procedure outlined in (a). Solid red curve in (d) is from numerical simulation [28 Sec. VII]. Experiments here and in other figures have sampling frequency ${\tau }_r/t_s=41$, trap stiffness $\kappa =42\mathrm{pN}/\mu \mathrm{m}$, scaled effective mass ${\delta }_g=0.8$, diffusion constant $D=0.12\mu {\mathrm{m}}^2/\mathrm{s}$, relaxation time ${\tau }_r=0.8\mathrm{ms}$, and bead diameter $4\mu \mathrm{m}$. Markers denote experimental means, and error bars the standard errors of the mean.

Figure 3

Performance of the information engines. (a) Output power of naive (red) and Bayesian (blue) information engines as a function of SNR. Hollow red markers denote output power at ${\alpha }^{*}=0$. (b) Difference of output work extraction rates for the Bayesian ($B$) and naive ($N$) engines scaled by the maximum rate (${\dot{F}}_{\max }=0.27$). The difference peaks at $\mathrm{SNR}={\mathrm{SNR}}_{\mathrm{c}}\approx 0.7$ (vertical dashed lines). Markers denote experimental means, solid curves the numerical simulations [28 Sec. VII]. Error bars denote (a) standard error of the mean and (b) propagated standard error of the mean from (a).