Stochastic resonance with weak monochromatic driving: gains above unity induced by high-frequency signals

We study the effects of a high-frequency (HF) signal on the response of a noisy bistable system to a low-frequency subthreshold sinusoidal signal. We show that, by conveniently choosing the ratio of the amplitude of the HF signal to its frequency, stochastic resonance gains greater than unity can be measured at the low-frequency value. Thus, the addition of the HF signal can entail an improvement in the detection of weak monochromatic signals. The results are explained in terms of an effective model and illustrated by means of numerical simulations. The phenomenon of stochastic resonance (SR) has been studied with growing interest during the last three decades, being found to be of relevance in a great variety of phenomena in physics, chemistry, and the life sciences [1]. Roughly speaking, SR consists in the amplification of a weak, time-dependent signal of interest by the concerted actions of noise and the nonlinearity of the system. Several quantifiers have been used to characterise the SR response of noisy systems in the presence of periodic signals. In particular, the nonmonotonic behaviour of the output signal-to-noise ratio (SNR) with the strength of the noise is a widely accepted signature of SR. In addition, a dimensionless quantity known as the SR gain is usually defined as the ratio of the output SNR over the input SNR. The SNR measures the “quality” of the signal, in terms of the ratio of its “coherent” (periodic) component over its “incoherent” (noisy) component. In turn, the SR gain compares the “qualities” of the output and the input signals. In general, obtaining high output SNRs and SR gains greater than unity would be desirable when using SR as an amplification mechanism. Analog [2,3] and numerical [4,5] simulations have shown that noisy bistable systems driven by subthreshold multifrequency forces can display SR gains greater than unity when the parameters of the problem are properly chosen. Moreover, in [6], a two-state model of SR has been used to explain these results analytically. By contrast, to the best of our knowledge, there is no evidence of SR gains greater than unity when a subthreshold monochromatic signal drives an isolated bistable system. However, these unusual large gains have been reported in the case of a suprathreshold sinusoidal signal for an isolated bistable system [7] and in the case of a subthreshold sinusoidal signal for coupled bistable systems [8]. As it has been already pointed out in the literature [7], an improvement of the response of a nonlinear system to a subthreshold signal of interest embedded in noise can be achieved better by lowering the threshold value rather than by increasing the noise strength. However, in most cases of practical interest, threshold lowering is usually a difficult task. Recently, it has been shown that the effect of a strong high-frequency (HF) monochromatic force on the overdamped dynamics of a Brownian particle in a bistable potential can be described in terms of an effective potential whose characteristics depend on the parameters of the HF field [9]. In particular, the barrier height of this effective potential is smaller than the original one. Thus, an HF force provides a mechanism to achieve an effective threshold lowering. As shown in [10] for a square-wave signal of interest, it is possible to take advantage of this effective threshold lowering to improve the SR gain values evaluated at the fundamental frequency of the square wave signal. More precisely, in [10], SR gains greater than unity are found for a wide range of values of the noise strength. However, as mentioned in the same reference, this positive effect of the HF force seems to be absent when the signal of interest is sinusoidal. It is pertinent to point out that the effect of HF fields on nonlinear stochastic systems is not necessarily positive (see, for instance, [11] for a ratchet