Stochastic resonant damping in a noisy monostable system: Theory and experiment

Usually in the presence of a background noise an increased effort put in controlling a system stabilizes its behavior. Rarely it is thought that an increased control of the system can lead to a looser response and, therefore, to a poorer performance. Strikingly there are many systems that show this weird behavior; examples can be drawn form physical, biological, and social systems. Until now no simple and general mechanism underlying such behaviors has been identified. Here we show that such a mechanism, named stochastic resonant damping, can be provided by the interplay between the background noise and the control exerted on the system. We experimentally verify our prediction on a physical model system based on a colloidal particle held in an oscillating optical potential. Our result adds a tool for the study of intrinsically noisy phenomena, joining the many constructive facets of noise identified in the past decades—for example, stochastic resonance, noise-induced activation, and Brownian ratchets.

Figure 1

Output system variance ${\sigma }_x^2$ as a function of the confinement effort $C$ ($D=1$, $A=1$) for harmonic (a) and colored Gaussian (b) parametric noise for various values of the frequency $f_0$: from bottom to top $f_0=100$, 31, 10, 3, 1 (thick curve), 0.3, 0.1, 0.03, 0.01 Hz. The thick line indicates the ${\sigma }_x^2$ minima. The shaded area represents the range of accessible output variance for a given value of the confinement effort. The additional axis shows the system intrinsic noise cutoff frequency $f_c$. Insets: a typical example of $x_0$ in time and frequency domain $\left(f_0=1\right)$.

Figure 2

(Color online) Intrinsic system output (gray filled area) and system output in the presence of a sinusoidal modulation ($f_0=1$, $D=1$, $A=1$) (red dashed curve) for three states of the system, indicated by dots in Fig. 1a for the modulation frequency $f_0=1Hz$: (i) $C=0.4$ $\left(f_c=0.064\text{Hz}\right)$, (ii) $C=2$ $\left(f_c=0.32\text{Hz}\right)$ corresponding to the absolute variance minimum, and (iii) $C=10$ $\left(f_c=1.6\text{Hz}\right)$. The output variance always exceeds the intrinsic variance.

Figure 3

(Color online) Experimental setup: $O_1$, $100\times$ objective; $O_2$, $40\times$ objective; ${\text{DM}}_1$, ${\text{DM}}_2$, and ${\text{DM}}_3$, dichroic mirrors; $M$, mirror; AOM/D, acousto-optic modulator/deflector; $L_1$ and $L_2$, optical system for conjugation of the output plane of the acousto-optical deflector and the entrance pupil of the objective $O_1$; PH, pinhole; BF, bandpass filter; QPD, quadrant photodetector.

Figure 4

(Color online) (a) Experimental and theoretical ${\sigma }_x^2$ as a function of the laser power in the presence of harmonic modulation of the trap center. From the bottom to the top the various sets of values correspond to $f_0=5$, 50, 500, 5000 Hz. The bar represents one standard deviation. The solid lines represent the theoretical prediction for the experimental parameters ($k=0.4\text{pN}/\mu \text{m}\text{mW}\times \text{trapping}\text{power}\left[\text{mW}\right]$, $\gamma =1.1\times {10}^{-8}\text{Ns}/\text{m}$, $A=100\text{nm}$). The disagreement between experimental data and theoretical results for the right end of the 5-Hz data set is observed because for such values of the confinement effort and intrinsic noise frequency the trapping potential is not harmonic anymore. (b) Data for the nonmodulated direction $y$. The two additional axes show the value of the confinement effort and the corresponding cutoff frequency of the intrinsic noise.

Figure 5

(Color online) Experimental and theoretical ${\sigma }_x^2$ as a function of the trapping power obtained with colored Gaussian forcing of the equilibrium position. Various sets of values are presented for $f_0=5$, 500, 50, 5 Hz. The bars represent one standard deviation. The solid lines represent the theoretical prediction for the experimental parameters ($k=0.7\text{pN}/\mu \text{m}\text{mW}\times \text{trapping}\text{power}\left[\text{mW}\right]$, $\gamma =1.9\times {10}^{-8}\text{Ns}/\text{m}$, $A=70\text{nm}$). The disagreement between experimental data and theoretical results for the right end of the 5-Hz data set is observed because for such values of the confinement effort and intrinsic noise frequency the trapping potential is not harmonic anymore. The two additional axes show the value of the confinement effort and the corresponding cutoff frequency of the intrinsic noise.

Figure 6

(Color online) Experimental ${\sigma }_x^2$ as a function of the noise amplitude. Various sets of values are presented for $A=55$ 70, 85 nm. The bars represent one standard deviation. The solid lines represent the theoretical prediction for the experimental parameters ($k=0.7\text{pN}/\mu \text{m}\text{mW}\times \text{trapping}\text{power}\left[\text{mW}\right]$, $\gamma =1.9\times {10}^{-8}\text{Ns}/\text{m}$). The two additional axes show the value of the confinement effort and the corresponding cutoff frequency of the intrinsic noise.