  A full-quantum approach is used to study quantum nonlinear properties of a
compound Michelson-Sagnac interferometer optomechanical system. The effective
Hamiltonian shows that both dissipative and dispersive couplings possess
imaginary- and real-Kerr nonlinearities. And unexpectedly, the nonlinearities
caused by the dissipative coupling have non-Hermitian Hamiltonian-like
properties. It can protect the quantum nature of the dispersive coupling beyond
the traditional dissipation of the system. This protection mechanism allows the
system to exhibit strong quantum nonlinear effects in the parameter region of
the hyperbolic function $J^2 = \Delta_c \Delta_e$. Moreover, we can obtain
strong anti-bunching effects whether in strong or weak coupling regimes with
the help of the dispersive and dissipative couplings jointly. It may provide a
new perspective to experimentally realize and study the strong quantum
nonlinear effects.
