  Feedback-based control of nano- and micromechanical resonators can enable the
study of macroscopic quantum phenomena and also sensitive force measurements.
Here, we demonstrate the feedback cooling of a low-loss and high-stress
macroscopic SiN membrane resonator close to its quantum ground state. We use
the microwave optomechanical platform, where the resonator is coupled to a
microwave cavity. The experiment utilizes a Josephson travelling wave
parametric amplifier, which is nearly quantum-limited in added noise, and is
important to mitigate resonator heating due to system noise in the feedback
loop. We reach a thermal phonon number as low as 1.6, which is limited
primarily by microwave-induced heating. We also discuss the sideband asymmetry
observed when a weak microwave tone for independent readout is applied in
addition to other tones used for the cooling. The asymmetry can be
qualitatively attributed to the quantum-mechanical imbalance between emission
and absorption. However, we find that the observed asymmetry is only partially
due to this quantum effect. In specific situations, the asymmetry is fully
dominated by a cavity Kerr effect under multitone irradiation.
