Nonlinear phase measurement using a coherent state in noisy environments

Abstract

Nonlinear phase measurement plays an important role in many dynamic processes. In this paper, we propose a single-mode measurement scheme for second-order nonlinear phase shifts that utilizes coherent states and homodyne measurement. The sensitivity at operating point φ = 0 is 1 / 4 N 3 / 2 with N photons on average and approaches the quantum Cramér–Rao bound. In addition, sensitivity with a small value of φ is discussed. For practical purposes, we analyze the effects of photon losses, thermal photon noise, and phase diffusion on sensitivities with φ = 0 as well as a small value of φ . The sensitivity still scales as 1 / N 3 / 2 in the presence of photon losses and thermal photon noise but degrades to the scaling of 1 / N with phase diffusion.