= Stochastic resonance
Stochastic resonance is an enhancement of a defined response to a weak signal by adding an appropriate amount of <noise> to a nonlinear system. The response measure and system matter: increasing noise is not a general method for improving every signal. Thresholds or switching between states can allow noise to reveal a modulation that would otherwise produce little observable output.
A primary optical experiment used a bidirectional ring <laser>. <Bruce McNamara>, <Kurt Wiesenfeld> and <Rajarshi Roy> periodically modulated the asymmetry between counter-rotating lasing modes and injected noise. They observed an improved <signal-to-noise ratio> relative to the case without externally injected noise.<Observation of Stochastic Resonance in a Ring Laser by Bruce McNamara and others>[Abstract, modulation apparatus and observed signal-to-noise improvement]{ref}
An original threshold model illustrates the competing effects without claiming to reproduce that laser. Let a detector report one when $A\cos(\omega t)+\eta>\theta$, with $0<A<\theta$ and independent Gaussian noise samples $\eta\sim N(0,\sigma^2)$. If $\Phi$ is the <standard normal distribution>'s <cumulative distribution function>, its crossing probability at signal phase $\phi$ is
$$
p(\phi)=1-\Phi\left(\frac{\theta-A\cos\phi}{\sigma}\right).
$$
The contrast between the most and least favorable phases is
$$
\Delta p=\Phi\left(\frac{\theta+A}{\sigma}\right)-\Phi\left(\frac{\theta-A}{\sigma}\right).
$$
It tends to zero both as $\sigma\to0$ and as $\sigma\to\infty$: without noise the subthreshold signal never crosses; overwhelming noise makes the two phases nearly indistinguishable.
For chosen $\theta=1$ and $A=0.5$, $\sigma=0.25$, $1$ and $5$ give contrasts about $0.0228$, $0.2417$ and $0.0781$. Moderate noise improves this particular phase-contrast statistic. These numbers are not laser data and are not themselves an output spectral <signal-to-noise ratio>.
A discriminating experiment holds the weak drive and response definition fixed while sweeping noise intensity, and includes a no-drive comparison. A larger output fluctuation is not enough: one must test whether the output better tracks the imposed signal. Switching rates, noise correlation and observation bandwidth can determine the useful regime in a dynamical system.
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