  This thesis focuses on the mathematical description and application of
nonlinear cavity optomechanical systems. The first part is concerned with
solving the dynamics of the standard nonlinear optomechanical Hamiltonian with
an additional time-dependent mechanical displacement and single-mode squeezing
term. The solution is based on identifying a Lie algebra that generates the
time-evolution of the system, which reduces the problem to considering a finite
set of coupled ordinary differential equations of real functions. The second
part applies the solutions of the extended optomechanical Hamiltonian to the
study of non-Gaussianity. We compute the non-Gaussian character of an
optomechanical state as a function of the parameters in the Hamiltonian, and
investigate the interplay between the non-Gaussianity, the strength of the
nonlinear coupling and the strength of the single-mode mechanical squeezing
term. We find that the strength and form of the nonlinear coupling strongly
impacts the non-Gaussianity, and that its relationship with the squeezing term
is highly complex. The third part concerns the use of nonlinear optomechanical
systems as quantum sensors. We derive a general expression of the quantum
Fisher information given the extended optomechanical Hamiltonian and
demonstrate its applicability through three concrete examples: estimating the
strength of a nonlinear light--matter coupling, the strength of a
time-modulated mechanical displacement, and the strength of a single-mode
mechanical squeezing parameter, all of which are modulated at resonance. In the
last Chapter of the thesis, we consider the estimation of a constant
gravitational acceleration with an optomechanical system. Our results suggest
that optomechanical systems could, in principle, be used as powerful quantum
sensors.
