On deep squeezing or cooling a parametric resonator using feedback

Source abstract

Abstract Here we analyze ways to achieve deep subthreshold squeezing or
cooling of fluctuations in a parametric resonator enhanced by a lock-in
amplifier feedback loop.
The main advance and novelty of this paper is that we analyzed the full impact
of this feedback on the response of the parametric resonator to an ac
drive or to additive noise.
Due to this feedback, the dynamics of the parametric resonator becomes more
complex and, in addition to a saddle-node bifurcation (SNB), a Hopf bifurcation (HB) at the instability threshold can also occur.
We calculate the phase-dependent amplification gain in the
response of the resonator to an added ac signal with three independent
techniques: approximately using the averaging method (AM) or the harmonic
balance method (HBM), and more precisely using Floquet theory (FT) and
Green's functions.
The HB was predicted by the AM, FT, and a modified version of the HBM that takes
into account quasi-periodicity.
In our analysis of fluctuations, we were able to calculate the
noise spectral density and the squeezing or cooling of fluctuations using
Fourier techniques and FT.
Deamplification and cooling occur near the HB, whereas deep squeezing and
heating occur near a SNB.
Furthermore, we show that there is an optimal amount of parametric pumping and
feedback to reach the strongest cooling.
The theoretical framework presented here is general and could be applied to a
wide range of physical implementations.