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Quantum-enhanced phase estimation paves the way to ultra-precision sensingand is of great realistic significance. In this paper we investigatetheoretically the estimation of a second-order nonlinear phase shift using acoherent state and parity measurement. A numerical expression is derived, theresolution and the sensitivity of parity signal are contrasted to linear phaseestimation protocol, and the signal visibility is analyzed. Additionally, byvirtue of phase-averaging approach to eliminate any hidden resources, we makean attempt at unveiling the low-down on the fundamental sensitivity limit fromthe quantum Fisher information. Finally, the effects of several realisticscenarios on the resolution and the sensitivity are studied, including photonloss, imperfect detector, and those which are a combination thereof.

Sub-Heisenberg-limited nonlinear phase estimation: Parity measurement approaches the quantum Cram\'er-Rao bound