  We propose a system composed of a qubit interacting with a quartic (undriven)
nonlinear oscillator (NLO) through a conditional displacement Hamiltonian. We
show that even a modest nonlinear perturbation in the NLO potential can enhance
and stabilize the quantum entanglement dynamically generated between the qubit
and the NLO. In absence of the nonlinearity the entanglement between the qubit
and the oscillator is known to periodically oscillate between 0 and 1, whereas
the nonlinearity suppresses the dynamical decay of entanglement once it is
generated. While the generation of entanglement is due to the superposition
principle combined with conditional displacements in the NLO, as noted in
several works before, the improvements in this entanglement is because of the
squeezing and other complex processes induced by two- and four-phonon
interactions. Finally, we have solved the Markovian master equation, and even
in this open case the system preserves the previous features and remain robust.
