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

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.