  Quantum-enhanced phase estimation paves the way to ultra-precision sensing
and is of great realistic significance. In this paper we investigate
theoretically the estimation of a second-order nonlinear phase shift using a
coherent state and parity measurement. A numerical expression is derived, the
resolution and the sensitivity of parity signal are contrasted to linear phase
estimation protocol, and the signal visibility is analyzed. Additionally, by
virtue of phase-averaging approach to eliminate any hidden resources, we make
an attempt at unveiling the low-down on the fundamental sensitivity limit from
the quantum Fisher information. Finally, the effects of several realistic
scenarios on the resolution and the sensitivity are studied, including photon
loss, imperfect detector, and those which are a combination thereof.
