  We investigate the quantum entanglement dynamics of undriven anharmonic
(nonlinear) oscillators with quartic potentials. We first consider the indirect
interaction between two such nonlinear oscillators mediated by a third, linear
oscillator and show that it leads to a time-varying entanglement of the
oscillators, the entanglement being strongly influenced by the nonlinear
oscillator dynamics. In the presence of dissipation, the role of nonlinearity
is strongly manifested in the steady state dynamics of the indirectly coupled
anharmonic oscillators. We further illustrate the effect of nonlinearities by
studying the coupling between an electromagnetic field in a cavity with one
movable mirror which is modeled as a nonlinear oscillator. For this case we
present a full analytical treatment, which is valid in a regime where both the
nonlinearity and the coupling due to radiation pressure is weak. We show that,
without the need of any conditional measurements on the cavity field, the state
of the movable mirror is non-classical as a result of the combined effect of
the intrinsic nonlinearity and the radiation pressure coupling. This
interaction is also shown to be responsible for squeezing the movable mirror's
position quadrature beyond the minimum uncertainty state even when the mirror
is initially prepared in its ground state.
