  We propose a feasible experimental scheme to improve the few-photon
optomechanical effects, including photon blockade and mechanical-Schrodinger
cat-state generation, as well as photon-phonon entanglement in a tripartite
microwave optomechanical circuit. The system under consideration is formed by a
single-Cooper-pair transistor, a microwave LC resonator, and a micromechanical
resonator. Our scheme is based on an additional higher-order (generalized)
nonlinear cross-Kerr type of coupling, linearly dependent on photon number
while quadratically dependent on mechanical phonon one, which can be realized
via adjusting the gate charge of the Cooper-pair transistor. We show, both
analytically and numerically, that the presence of both cross-Kerr and
generalized cross-Kerr nonlinearities not only may give rise to the enhancement
of one- and two-photon blockades as well as photon induced tunneling but can
also provide more controllability over them. Furthermore, it is shown that in
the regime of zero optomechanical coupling, with the aid of generalized
cross-Kerr nonlinearity, one can generate multi-components mechanical
superposition states which exhibit robustness against system dissipations. We
also study the steady-state entanglement between the microwave and mechanical
modes, the results of which signify the role of generalized cross-Kerr
nonlinearity in enhancing the entanglement in the regime of large-red detuning.
The proposed generalized cross-Kerr optomechanical system can be found
potential applications in microwave quantum sensing, quantum telecommunication,
and quantum information protocols.
