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

TL;DR

This paper investigates methods to achieve deep subthreshold squeezing and cooling of a parametric resonator using feedback, analyzing complex dynamics and bifurcations with multiple theoretical approaches.

Contribution

It introduces a comprehensive analysis of feedback-enhanced parametric squeezing and cooling, including new insights into bifurcation behavior and fluctuation suppression.

Findings

Strong squeezing and cooling are achievable near bifurcation points.

Multiple theoretical methods provide consistent predictions of bifurcation and gain.

Feedback modifies the dynamics, enabling enhanced fluctuation suppression.

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.