  We study the fundamental sensitivity that can be achieved with an ideal
optomechanical system in the nonlinear regime for measurements of
time-dependent gravitational fields. Using recently developed methods to solve
the dynamics of a nonlinear optomechanical system with a time-dependent
Hamiltonian, we compute the quantum Fisher information for linear displacements
of the mechanical element due to gravity. We demonstrate that the sensitivity
can not only be further enhanced by injecting squeezed states of the cavity
field, but also by modulating the light--matter coupling of the optomechanical
system. We specifically apply our results to the measurement of gravitational
fields from small oscillating masses, where we show that, in principle, the
gravitational field of an oscillating nano-gram mass can be detected based on
experimental parameters that will likely be accessible in the near-term future.
Finally, we identify the experimental parameter regime necessary for
gravitational wave detection with a quantum optomechanical sensor.
