Gain and noise spectral density in a parametric amplifier with added white noise: Theory and experiment

Abstract

In this paper, we discuss the behavior of a linear classical parametric amplifier in the presence of white noise and give theoretical estimates of the noise spectral density based on approximate Green’s functions obtained by using averaging techniques. To validate our theory, we compare the analytical results with experimental data from an analog circuit and with numerical simulations of the model’s stochastic differential equations. The experimental data were accurately described by our model. Moreover, we noticed spectral components in the output signal of the amplifier, which are due to noisy precursors of instability. The position, width, and magnitude of these components are in agreement with the noise spectral density obtained by the theory proposed here.

Amplification in parametrically-driven resonators near instability based on Floquet theory and Green’s functions

“…Furthermore, if one is not very close to the instability threshold, the Floquet exponents could be a complex conjugate pair in the case of a one-degree-of-freedom parametric amplifier as one can verify from the Liouville's formula [14]. This fact results in split peaks such as predicted theoretically and observed experimentally in [15]. These split peaks have also been observed in parametric resonators with very high quality factors [16].…”

“…Here we used the noise level D = 3.08 × 10 −8 (in dimensionless units) and the quality factor Q = 65. These values were obtained from the electronic circuit implementation of a parametric oscillator given in [15]. In this case, x(t) has dimensions of voltage (measured in units of V ) and the noise level has dimensions of squared voltage over frequency (measured in units of V 2 /Hz).…”

“…To transform our dimensionless D to the experimental noise level, we divide it by ω 0 the natural angular frequency of the resonator in units of rad/s and multiply it by V 2 . The averaging method results are obtained from the approximations to the Green's functions Fourier transforms The first-order approximation to the frequency-domain response is given in [15]. The second-order Figure 1.…”

“…. The expression given in equation ( 26) is equivalent to the one obtained in [15]. While the previous expression is approximate, based on the 1st-order averaging method, and specific to the one-degree-of-freedom parametric resonator with added noise, the one proposed here is more general and exact, since it is a direct application of Floquet theory.…”

“…We used this fact to obtain a more intuitive expression for the response of the parametric resonator to an added ac signal or white noise. Here, we extended our Fourier analysis as presented in [15], based on the approximate averaging method, to the, in principle, exact Floquet theory result. Furthermore, the results for the response of the resonator to the drive and the NSD seem more intuitive physically than the ones provided in the literature.…”