  In this paper, we propose an experimentally viable scheme to enhance the
sensitivity of force detection in a hybrid optomechanical setup assisted by
squeezed vacuum injection, beyond the standard quantum limit (SQL). The scheme
is based on a combination of the coherent quantum noise cancellation (CQNC)
strategy with a variational homodyne detection of the cavity output spectrum in
which the phase of the local oscillator is optimized. In CQNC, realizing a
negative-mass oscillator in the system leads to exact cancellation of the
backaction noise from the mechanics due to destructive quantum interference.
Squeezed vacuum injection enhances this cancellation and allows sub-SQL
sensitivity to be reached in a wide frequency band and at much lower input
laser powers. We show here that the adoption of variational homodyne readout
enables us to enhance this noise cancellation up to $40 ~\mathrm{dB}$ compared
to the standard case of detection of the optical output phase quadrature,
leading to a remarkable force sensitivity of the order of $10^{-19}
\mathrm{N}/\sqrt{\mathrm{Hz}}$, around 2-order enhancement compared to the
standard case. Moreover, we show that at nonzero cavity detuning, the signal
response can be amplified at a level three to five times larger than that in
the standard case without variational homodyne readout, improving the
signal-to-noise-ratio (SNR). Finally, the variational readout CQNC developed in
this paper may be applied to other optomechanical-like platforms such as
levitated systems and multimode optomechanical arrays or crystals as well as
Josephson-based optomechanical systems.
