Deep squeezing or cooling the fluctuations of a parametric resonator using feedback

Abstract

Here we analyze ways to achieve deep subthreshold parametric squeezing or cooling of a single degree-of-freedom parametric resonator enhanced by a lock-in amplifier feedback loop. Due to the feedback, the dynamics of the parametric resonator becomes more complex and a Hopf bifurcation at the instability threshold can occur. Initially, we calculate the phase-dependent gain of parametric amplification with feedback of an added ac signal. In one approach, we obtain the amplification gain approximately using two independent approaches: the averaging method and the harmonic balance method. We also obtain this gain more exactly using Floquet theory and Green's functions methods. The Hopf bifurcation was predicted by the harmonic balance method and by Floquet theory, but not by the averaging method. In our analysis of fluctuations, we Fourier analyze the response of the parametric resonator with feedback to an added white noise. We were able to calculate, in addition to the noise spectral density, the squeezing of fluctuations in this resonator with feedback. Very strong squeezing or cooling can occur. Deamplification and cooling occur near the Hopf bifurcation, whereas squeezing occurs near a saddle-node bifurcation.

Figures (8)

• Figure 1: Parametric instability threshold line for the SDOF parametric resonator with LIA feedback as described by Eq. \ref{['pa_lockin_feedback_amp2']} with $F_s=0$. A The threshold line to instability occurs at a saddle-node bifurcation (when one real Floquet multiplier becomes equal to 1). Comparison between the FT prediction, the HBM prediction (from the characteristic equation $\det M=0$ in Eq. \ref{['eq:detM']}), and the averaging prediction (the positive root of Eq. \ref{['threshold']}). B The threshold line to instability occurs at a Hopf bifurcation (when the module of the complex conjugate pair of Floquet multipliers becomes equal to 1). Comparison between FT prediction and the HBM prediction, which is obtained from solving the algebraic system from Eq. \ref{['Fp_delta_system']}.
• Figure 2: Plots of the Floquet multipliers as the pump amplitude is varied for the parameters shown at the top of the figure. The vertical axis was compressed for better visibility. The dashed line corresponds to the instability threshold at $|\mu|=1$. The '$\times$' symbols represent the Floquet multipliers at $F_p=0$. A The pump amplitude $F_p$ is varied from $0$ to $0.002$. A saddle-node bifurcation occurs at $F_p\approx0.002$ when the real FM reaches the values $1$ while the complex FMs still have magnitude less than 1. B The pump amplitude $F_p$ is varied from $0$ to $-0.042$. A Hopf bifurcation occurs at $F_p\approx -0.042$ when the pair of complex FMs reaches the unit circle while the real FM still has magnitude less than 1.
• Figure 3: A A time-series transient of $x(t)$ obtained from numerical integration of Eqs. \ref{['pa_lockin_feedback_amp2']} with parameters set near a Hopf bifurcation point. The envelopes (red lines) are obtained from the averaging approximation by integrating Eqs. \ref{['uvz_avg']}. The envelopes are given by $\pm \sqrt{u^2(t)+v^2(t)}$. B Fourier transform of the stationary part of the time series shown in panel A. The two peaks are a signature of quasi-periodic behaviour that arises near the onset of a Hopf bifurcation. The dashed lines were obtained from the harmonic balance method by solving Eq. \ref{['Fp_delta_system']}.
• Figure 4: Gain as a function of phase for the same parametric amplifier with feedback of Fig. fig:transient with parameters set near the onset of a saddle-node bifurcation. A We plot the normalized cyclo-stationary response $x(t)/x_0$ as a function of phase $\varphi$, where $x_0$ is the amplitude of oscillations of the forced harmonic resonator (when $F_p=\eta=0$). There is a slight detuning between $\omega$ and $\omega_s$ so that the phase is swept very slowly, almost quasi-statically. The envelopes are obtained from Eq. \ref{['eq:envelopes']}, which are based on FT. B We use the positive envelope from A to obtain the quasidegenerate gain in decibels. The FT degenerate gain is obtained in Eq. \ref{['gain_FT_fb']}. The analytical HBM gain is given in Eq. \ref{['gain_HBM']} with the help of Eq. \ref{['eq:A_x']}. The horizontal lines are the HBM approximations for minimum and maximum gains obtained from Eq. \ref{['G_minG_max']}.
• Figure 5: Gain as a function of phase for the SDOF parametric amplifier with LIA feedback whose dynamics is given in Eq. \ref{['pa_lockin_feedback_amp2']} with parameters set near the onset of a Hopf bifurcation. A The response of the SDOF parametric resonator with feedback is far reduced compared with the amplitude of the harmonic oscillator response. B There is attenuation in all phases. This indicates that with these parameters the parametric resonator with the proposed feedback scheme leads to cooling.